Although this hadith seems to be missing a narrator (unknown narrator), it doesn't.
That narrator is called سعد بن عبيدة Sa'd bin Obaidah, he was clearly named in other versions of the hadith you quoted.
The version you narrated of that hadith is this:
Narrated: (The translation doesn't mention the list of narrators)
Abu 'Abdur-Rahman and Hibban bin 'Atiyya had a dispute. Abu 'Abdur-Rahman said to Hibban, "You know what made your companions (i.e. 'Ali) dare to shed blood." Hibban said, "Come on! What is that?" 'Abdur-Rahman said, "Something I heard him saying." The other said, "What was it?" 'AbdurRahman said, "'Ali said, Allah's Apostle sent for me, Az-Zubair and Abu Marthad, and all of us were cavalry men, and said, 'Proceed to Raudat-Hajj (Abu Salama said that Abu 'Awana called it like this, i.e., Hajj where there is a woman carrying a letter from Hatib bin Abi Balta'a to the pagans (of Mecca). So bring that letter to me.' So we proceeded riding on our horses till we overtook her at the same place of which Allah's Apostle had told us. She was traveling on her camel. In that letter Hatib had written to the Meccans about the proposed attached of Allah's Apostle against them. We asked her, "Where is the letter which is with you?' She replied, 'I haven't got any letter.' So we made her camel kneel down and searched her luggage, but we did not find anything. My two companions said, 'We do not think that she has got a letter.' I said, 'We know that Allah's Apostle has not told a lie.'" Then Ali took an oath saying, "By Him by Whom one should swear! You shall either bring out the letter or we shall strip off your clothes." She then stretched out her hand for her girdle (round her waist) and brought out the paper (letter). They took the letter to Allah's Apostle. Umar said, "O Allah's Apostle! (Hatib) has betrayed Allah, His Apostle and the believers; let me chop off his neck!" Allah's Apostle said, "O Hatib! What obliged you to do what you have done?" Hatib replied, "O Allah's Apostle! Why (for what reason) should I not believe in Allah and His Apostle? But I intended to do the (Mecca) people a favor by virtue of which my family and property may be protected as there is none of your companions but has some of his people (relatives) whom Allah urges to protect his family and property." The Prophet said, "He has said the truth; therefore, do not say anything to him except good." Umar again said, "O Allah's Apostle! He has betrayed Allah, His Apostle and the believers; let me chop his neck off!" The Prophet said, "Isn't he from those who fought the battle of Badr? And what do you know, Allah might have looked at them (Badr warriors) and said (to them), 'Do what you like, for I have granted you Paradise?' " On that, Umar's eyes became flooded with tears and he said, "Allah and His Apostle know best."
حَدَّثَنَا مُوسَى بْنُ إِسْمَاعِيلَ، حَدَّثَنَا أَبُو عَوَانَةَ، عَنْ حُصَيْنٍ، عَنْ فُلاَنٍ، قَالَ تَنَازَعَ أَبُو عَبْدِ الرَّحْمَنِ وَحِبَّانُ بْنُ عَطِيَّةَ فَقَالَ أَبُو عَبْدِ الرَّحْمَنِ لِحِبَّانَ لَقَدْ عَلِمْتُ الَّذِي جَرَّأَ صَاحِبَكَ عَلَى الدِّمَاءِ يَعْنِي عَلِيًّا. قَالَ مَا هُوَ لاَ أَبَا لَكَ قَالَ شَىْءٌ سَمِعْتُهُ يَقُولُهُ. قَالَ مَا هُوَ قَالَ بَعَثَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم وَالزُّبَيْرَ وَأَبَا مَرْثَدٍ وَكُلُّنَا فَارِسٌ قَالَ " انْطَلِقُوا حَتَّى تَأْتُوا رَوْضَةَ حَاجٍ ـ قَالَ أَبُو سَلَمَةَ هَكَذَا قَالَ أَبُو عَوَانَةَ حَاجٍ ـ فَإِنَّ فِيهَا امْرَأَةً مَعَهَا صَحِيفَةٌ مِنْ حَاطِبِ بْنِ أَبِي بَلْتَعَةَ إِلَى الْمُشْرِكِينَ فَأْتُونِي بِهَا ". فَانْطَلَقْنَا عَلَى أَفْرَاسِنَا حَتَّى أَدْرَكْنَاهَا حَيْثُ قَالَ لَنَا رَسُولُ اللَّهِ صلى الله عليه وسلم تَسِيرُ عَلَى بَعِيرٍ لَهَا، وَكَانَ كَتَبَ إِلَى أَهْلِ مَكَّةَ بِمَسِيرِ رَسُولِ اللَّهِ صلى الله عليه وسلم إِلَيْهِمْ. فَقُلْنَا أَيْنَ الْكِتَابُ الَّذِي مَعَكِ قَالَتْ مَا مَعِي كِتَابٌ. فَأَنَخْنَا بِهَا بَعِيرَهَا، فَابْتَغَيْنَا فِي رَحْلِهَا فَمَا وَجَدْنَا شَيْئًا. فَقَالَ صَاحِبِي مَا نَرَى مَعَهَا كِتَابًا. قَالَ فَقُلْتُ لَقَدْ عَلِمْنَا مَا كَذَبَ رَسُولُ اللَّهِ صلى الله عليه وسلم ثُمَّ حَلَفَ عَلِيٌّ وَالَّذِي يُحْلَفُ بِهِ لَتُخْرِجِنَّ الْكِتَابَ أَوْ لأُجَرِّدَنَّكِ. فَأَهْوَتْ إِلَى حُجْزَتِهَا وَهْىَ مُحْتَجِزَةٌ بِكِسَاءٍ فَأَخْرَجَتِ الصَّحِيفَةَ، فَأَتَوْا بِهَا رَسُولَ اللَّهِ صلى الله عليه وسلم فَقَالَ عُمَرُ يَا رَسُولَ اللَّهِ قَدْ خَانَ اللَّهَ وَرَسُولَهُ وَالْمُؤْمِنِينَ. دَعْنِي فَأَضْرِبَ عُنُقَهُ. فَقَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم " يَا حَاطِبُ مَا حَمَلَكَ عَلَى مَا صَنَعْتَ ". قَالَ يَا رَسُولَ اللَّهِ مَالِي أَنْ لاَ أَكُونَ مُؤْمِنًا بِاللَّهِ وَرَسُولِهِ، وَلَكِنِّي أَرَدْتُ أَنْ يَكُونَ لِي عِنْدَ الْقَوْمِ يَدٌ، يُدْفَعُ بِهَا عَنْ أَهْلِي وَمَالِي، وَلَيْسَ مِنْ أَصْحَابِكَ أَحَدٌ إِلاَّ لَهُ هُنَالِكَ مِنْ قَوْمِهِ مَنْ يَدْفَعُ اللَّهُ بِهِ عَنْ أَهْلِهِ وَمَالِهِ. قَالَ " صَدَقَ، لاَ تَقُولُوا لَهُ إِلاَّ خَيْرًا ". قَالَ فَعَادَ عُمَرُ فَقَالَ يَا رَسُولُ اللَّهِ قَدْ خَانَ اللَّهَ وَرَسُولَهُ وَالْمُؤْمِنِينَ، دَعْنِي فَلأَضْرِبَ عُنُقَهُ. قَالَ " أَوَلَيْسَ مِنْ أَهْلِ بَدْرٍ، وَمَا يُدْرِيكَ لَعَلَّ اللَّهَ اطَّلَعَ عَلَيْهِمْ فَقَالَ اعْمَلُوا مَا شِئْتُمْ فَقَدْ أَوْجَبْتُ لَكُمُ الْجَنَّةَ ". فَاغْرَوْرَقَتْ عَيْنَاهُ فَقَالَ اللَّهُ وَرَسُولُهُ أَعْلَمُ.
[Sahih Bukhari]
There are other versions of the hadith both in sahih Bukhari and in other hadith books, what we care about now is Sahih bukhari itself.
Here are the other versions:
Narrated 'Ali: (Also the translation doesn't list narrators)
Allah's Apostle sent me, Az-Zubair bin Al-Awwam and Abu Marthad Al-Ghanawi, and all of us were horsemen, and he said, "Proceed till you reach Rawdat Khakh, where there is a woman from the pagans carrying a letter sent by Hatib bin Abi Balta'a to the pagans (of Mecca)." So we overtook her while she was proceeding on her camel at the same place as Allah's Apostle told us. We said (to her) "Where is the letter which is with you?" She said, "I have no letter with me." So we made her camel kneel down and searched her mount (baggage etc) but could not find anything. My two companions said, "We do not see any letter." I said, "I know that Allah's Apostle did not tell a lie. By Allah, if you (the lady) do not bring out the letter, I will strip you of your clothes' When she noticed that I was serious, she put her hand into the knot of her waist sheet, for she was tying a sheet round herself, and brought out the letter. So we proceeded to Allah's Apostle with the letter. The Prophet said (to Habib), "What made you o what you have done, O Hatib?" Hatib replied, "I have done nothing except that I believe in Allah and His Apostle, and I have not changed or altered (my religion). But I wanted to do the favor to the people (pagans of Mecca) through which Allah might protect my family and my property, as there is none among your companions but has someone in Mecca through whom Allah protects his property (against harm). The Prophet said, "Habib has told you the truth, so do not say to him (anything) but good." 'Umar bin Al-Khattab said, "Verily he has betrayed Allah, His Apostle, and the believers! Allow me to chop his neck off!" The Prophet said, "O 'Umar! What do you know; perhaps Allah looked upon the Badr warriors and said, 'Do whatever you like, for I have ordained that you will be in Paradise.'" On that 'Umar wept and said, "Allah and His Apostle know best."
حَدَّثَنَا يُوسُفُ بْنُ بُهْلُولٍ، حَدَّثَنَا ابْنُ إِدْرِيسَ، قَالَ حَدَّثَنِي حُصَيْنُ بْنُ عَبْدِ الرَّحْمَنِ، عَنْ سَعْدِ بْنِ عُبَيْدَةَ، عَنْ أَبِي عَبْدِ الرَّحْمَنِ السُّلَمِيِّ، عَنْ عَلِيٍّ ـ رضى الله عنه ـ قَالَ بَعَثَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم وَالزُّبَيْرَ بْنَ الْعَوَّامِ وَأَبَا مَرْثَدٍ الْغَنَوِيَّ وَكُلُّنَا فَارِسٌ فَقَالَ " انْطَلِقُوا حَتَّى تَأْتُوا رَوْضَةَ خَاخٍ، فَإِنَّ بِهَا امْرَأَةً مِنَ الْمُشْرِكِينَ مَعَهَا صَحِيفَةٌ مِنْ حَاطِبِ بْنِ أَبِي بَلْتَعَةَ إِلَى الْمُشْرِكِينَ ". قَالَ فَأَدْرَكْنَاهَا تَسِيرُ عَلَى جَمَلٍ لَهَا حَيْثُ قَالَ لَنَا رَسُولُ اللَّهِ صلى الله عليه وسلم قَالَ قُلْنَا أَيْنَ الْكِتَابُ الَّذِي مَعَكِ قَالَتْ مَا مَعِي كِتَابٌ. فَأَنَخْنَا بِهَا، فَابْتَغَيْنَا فِي رَحْلِهَا فَمَا وَجَدْنَا شَيْئًا، قَالَ صَاحِبَاىَ مَا نَرَى كِتَابًا. قَالَ قُلْتُ لَقَدْ عَلِمْتُ مَا كَذَبَ رَسُولُ اللَّهِ صلى الله عليه وسلم وَالَّذِي يُحْلَفُ بِهِ لَتُخْرِجِنَّ الْكِتَابَ أَوْ لأُجَرِّدَنَّكِ. قَالَ فَلَمَّا رَأَتِ الْجِدَّ مِنِّي أَهْوَتْ بِيَدِهَا إِلَى حُجْزَتِهَا وَهْىَ مُحْتَجِزَةٌ بِكِسَاءٍ فَأَخْرَجَتِ الْكِتَابَ ـ قَالَ ـ فَانْطَلَقْنَا بِهِ إِلَى رَسُولِ اللَّهِ صلى الله عليه وسلم فَقَالَ " مَا حَمَلَكَ يَا حَاطِبُ عَلَى مَا صَنَعْتَ ". قَالَ مَا بِي إِلاَّ أَنْ أَكُونَ مُؤْمِنًا بِاللَّهِ وَرَسُولِهِ، وَمَا غَيَّرْتُ وَلاَ بَدَّلْتُ، أَرَدْتُ أَنْ تَكُونَ لِي عِنْدَ الْقَوْمِ يَدٌ يَدْفَعُ اللَّهُ بِهَا عَنْ أَهْلِي وَمَالِي، وَلَيْسَ مِنْ أَصْحَابِكَ هُنَاكَ إِلاَّ وَلَهُ مَنْ يَدْفَعُ اللَّهُ بِهِ عَنْ أَهْلِهِ وَمَالِهِ. قَالَ " صَدَقَ فَلاَ تَقُولُوا لَهُ إِلاَّ خَيْرًا ". قَالَ فَقَالَ عُمَرُ بْنُ الْخَطَّابِ إِنَّهُ قَدْ خَانَ اللَّهَ وَرَسُولَهُ وَالْمُؤْمِنِينَ، فَدَعْنِي فَأَضْرِبَ عُنُقَهُ. قَالَ فَقَالَ " يَا عُمَرُ وَمَا يُدْرِيكَ لَعَلَّ اللَّهَ قَدِ اطَّلَعَ عَلَى أَهْلِ بَدْرٍ فَقَالَ اعْمَلُوا مَا شِئْتُمْ فَقَدْ وَجَبَتْ لَكُمُ الْجَنَّةُ ". قَالَ فَدَمَعَتْ عَيْنَا عُمَرَ وَقَالَ اللَّهُ وَرَسُولُهُ أَعْلَمُ.
[Sahih Bukhari]
And this:
Narrated 'Ali: (Again, the translation doesn't list the narrators)
Allah's Apostle sent me, Abu Marthad and Az-Zubair, and all of us were riding horses, and said, "Go till you reach Raudat-Khakh where there is a pagan woman carrying a letter from Hatib bin Abi Balta' a to the pagans of Mecca." So we found her riding her camel at the place which Allah's Apostle had mentioned. We said (to her),"(Give us) the letter." She said, "I have no letter." Then we made her camel kneel down and we searched her, but we found no letter. Then we said, "Allah's Apostle had not told us a lie, certainly. Take out the letter, otherwise we will strip you naked." When she saw that we were determined, she put her hand below her waist belt, for she had tied her cloak round her waist, and she took out the letter, and we brought her to Allah's Apostle Then 'Umar said, "O Allah's Apostle! (This Hatib) has betrayed Allah, His Apostle and the believers! Let me cut off his neck!" The Prophet asked Hatib, "What made you do this?" Hatib said, "By Allah, I did not intend to give up my belief in Allah and His Apostle but I wanted to have some influence among the (Mecca) people so that through it, Allah might protect my family and property. There is none of your companions but has some of his relatives there through whom Allah protects his family and property." The Prophet said, "He has spoken the truth; do no say to him but good." 'Umar said, "He as betrayed Allah, His Apostle and the faithful believers. Let me cut off his neck!" The Prophet said, "Is he not one of the Badr warriors? May be Allah looked at the Badr warriors and said, 'Do whatever you like, as I have granted Paradise to you, or said, 'I have forgiven you."' On this, tears came out of 'Umar's eyes, and he said, "Allah and His Apostle know better."
حَدَّثَنِي إِسْحَاقُ بْنُ إِبْرَاهِيمَ، أَخْبَرَنَا عَبْدُ اللَّهِ بْنُ إِدْرِيسَ، قَالَ سَمِعْتُ حُصَيْنَ بْنَ عَبْدِ الرَّحْمَنِ، عَنْ سَعْدِ بْنِ عُبَيْدَةَ، عَنْ أَبِي عَبْدِ الرَّحْمَنِ السُّلَمِيِّ، عَنْ عَلِيٍّ ـ رضى الله عنه ـ قَالَ بَعَثَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم وَأَبَا مَرْثَدٍ وَالزُّبَيْرَ وَكُلُّنَا فَارِسٌ قَالَ " انْطَلِقُوا حَتَّى تَأْتُوا رَوْضَةَ خَاخٍ، فَإِنَّ بِهَا امْرَأَةً مِنَ الْمُشْرِكِينَ، مَعَهَا كِتَابٌ مِنْ حَاطِبِ بْنِ أَبِي بَلْتَعَةَ إِلَى الْمُشْرِكِينَ ". فَأَدْرَكْنَاهَا تَسِيرُ عَلَى بَعِيرٍ لَهَا حَيْثُ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم فَقُلْنَا الْكِتَابُ. فَقَالَتْ مَا مَعَنَا كِتَابٌ. فَأَنَخْنَاهَا فَالْتَمَسْنَا فَلَمْ نَرَ كِتَابًا، فَقُلْنَا مَا كَذَبَ رَسُولُ اللَّهِ صلى الله عليه وسلم، لَتُخْرِجِنَّ الْكِتَابَ أَوْ لَنُجَرِّدَنَّكِ. فَلَمَّا رَأَتِ الْجِدَّ أَهْوَتْ إِلَى حُجْزَتِهَا وَهْىَ مُحْتَجِزَةٌ بِكِسَاءٍ فَأَخْرَجَتْهُ، فَانْطَلَقْنَا بِهَا إِلَى رَسُولِ اللَّهِ صلى الله عليه وسلم فَقَالَ عُمَرُ يَا رَسُولَ اللَّهِ، قَدْ خَانَ اللَّهَ وَرَسُولَهُ وَالْمُؤْمِنِينَ، فَدَعْنِي فَلأَضْرِبْ عُنُقَهُ. فَقَالَ النَّبِيُّ صلى الله عليه وسلم " مَا حَمَلَكَ عَلَى مَا صَنَعْتَ ". قَالَ حَاطِبٌ وَاللَّهِ مَا بِي أَنْ لاَ أَكُونَ مُؤْمِنًا بِاللَّهِ وَرَسُولِهِ صلى الله عليه وسلم أَرَدْتُ أَنْ يَكُونَ لِي عِنْدَ الْقَوْمِ يَدٌ يَدْفَعُ اللَّهُ بِهَا عَنْ أَهْلِي وَمَالِي، وَلَيْسَ أَحَدٌ مِنْ أَصْحَابِكَ إِلاَّ لَهُ هُنَاكَ مِنْ عَشِيرَتِهِ مَنْ يَدْفَعُ اللَّهُ بِهِ عَنْ أَهْلِهِ وَمَالِهِ. فَقَالَ النَّبِيُّ صلى الله عليه وسلم " صَدَقَ، وَلاَ تَقُولُوا لَهُ إِلاَّ خَيْرًا ". فَقَالَ عُمَرُ إِنَّهُ قَدْ خَانَ اللَّهَ وَرَسُولَهُ وَالْمُؤْمِنِينَ، فَدَعْنِي فَلأَضْرِبَ عُنُقَهُ. فَقَالَ " أَلَيْسَ مِنْ أَهْلِ بَدْرٍ ". فَقَالَ " لَعَلَّ اللَّهَ اطَّلَعَ إِلَى أَهْلِ بَدْرٍ فَقَالَ اعْمَلُوا مَا شِئْتُمْ فَقَدْ وَجَبَتْ لَكُمُ الْجَنَّةُ، أَوْ فَقَدْ غَفَرْتُ لَكُمْ ". فَدَمَعَتْ عَيْنَا عُمَرَ وَقَالَ اللَّهُ وَرَسُولُهُ أَعْلَمُ.
[Sahih Bukhari]
If you compared the Arabic list of narrators, you can see that سعد بن عبيدة in the two last versions comes in place of that Unknown narrator in the first version (yours).
The questions now is: If Imam Bukhari knew the narrator was سعد بن عبيدة, why didn't he mention that in the first version?
The answer is that Imam Bukhari heard the two last versions from people who mentioned the names of that narrator, and heard the first version from someone who didn't knew the name of that narrator; Imam Bukhari surely knew who the narrator was, but he never changed the narration, he recorded it just as he heard it from the one who told it to him.
That strongly proves the authenticity and honesty of (great) Imam Bukhari رحمه الله تعالى (May Allah be pleased with him).
what makes you think he witnessed it?
There is absence of narrator's name as you can see.