There are 2 parts to the neck - رقبة / قفی which is the back of the neck, and the حلقوم which is the front.
Wiping the front is bid'ah - the Prophet (Saw) never did this
Wiping the back WAS done by the Prophet (Saw), and the regularity is what is disputed.
Those ulema who say the Prophet (Saw) did it reguarly, call it Sunnah (an act worthy of reward.).
Those ulema who say the Prophet (Saw) did it occasionally, call it mustahabb or adab. (a good thing to do)
Those ulema who say it is not established from strong hadiths call it mubah (allowed).
NOBODY i know of calls wiping the back of the neck a bidah.
Al kasani - the author of the Hanafi fiqh book Badaius sanaai, (died 587hijri) says:
الْفَرْقُ بَيْنَ السُّنَّةِ، وَالْأَدَبِ أَنَّ السُّنَّةَ مَا، وَاظَبَ عَلَيْهِ رَسُولُ اللَّهِ - صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ - وَلَمْ يَتْرُكْهُ إلَّا مَرَّةً، أَوْ مَرَّتَيْنِ لِمَعْنًى مِنْ الْمَعَانِي،، وَالْأَدَبُ مَا فَعَلَهُ مَرَّةً، أَوْ مَرَّتَيْنِ، وَلَمْ يُوَاظِبْ عَلَيْهِ
the difference between a sunnah and an adab (which includes mustahabb and mubah) is that the actions the Prophet (Saw) did regualrly, and left out only once or twice due to a particular reason, are called SUNNAH. As for an ADAB, this is what the Prophet (Saw) did maybe once or twice, and didnt keep on regularly doing it.
As for all the hadiths being weak, this is wrong. Imam Muslim narrates a hadith about wiping the back of the neck:
وَحَدَّثَنِي إِسْحَاقُ بْنُ مُوسَى الأَنْصَارِيُّ، حَدَّثَنَا مَعْنٌ، حَدَّثَنَا مَالِكُ بْنُ أَنَسٍ، عَنْ عَمْرِو بْنِ يَحْيَى، بِهَذَا الإِسْنَادِ وَقَالَ مَضْمَضَ وَاسْتَنْثَرَ ثَلاَثًا . وَلَمْ يَقُلْ مِنْ كَفٍّ وَاحِدَةٍ . وَزَادَ بَعْدَ قَوْلِهِ فَأَقْبَلَ بِهِمَا وَأَدْبَرَ بَدَأَ بِمُقَدَّمِ رَأْسِهِ ثُمَّ ذَهَبَ بِهِمَا إِلَى قَفَاهُ ثُمَّ رَدَّهُمَا حَتَّى رَجَعَ إِلَى الْمَكَانِ الَّذِي بَدَأَ مِنْهُ وَغَسَلَ رِجْلَيْهِ .
Malik b. Anas narrated it from 'Amr b. Yahya with the same chain of transmitters, transmitters and mentioned the rinsing (of mouth) and snuffing (of water into the nostrils) three times, but he did not mention" from one palm," and made this addition:
He moved them (his hands) for wiping to the front of his head and then the nape of his neck, then bringing them back till he reached the place from which he had begun, after which he washed his feet.
it is also mentioned in Tirmidhi.
حَدَّثَنَا إِسْحَاقُ بْنُ مُوسَى الأَنْصَارِيُّ، حَدَّثَنَا مَعْنُ بْنُ عِيسَى الْقَزَّازُ، حَدَّثَنَا مَالِكُ بْنُ أَنَسٍ، عَنْ عَمْرِو بْنِ يَحْيَى، عَنْ أَبِيهِ، عَنْ عَبْدِ اللَّهِ بْنِ زَيْدٍ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم مَسَحَ رَأْسَهُ بِيَدَيْهِ فَأَقْبَلَ بِهِمَا وَأَدْبَرَ بَدَأَ بِمُقَدَّمِ رَأْسِهِ ثُمَّ ذَهَبَ بِهِمَا إِلَى قَفَاهُ ثُمَّ رَدَّهُمَا حَتَّى رَجَعَ إِلَى الْمَكَانِ الَّذِي بَدَأَ مِنْهُ ثُمَّ غَسَلَ رِجْلَيْهِ . قَالَ أَبُو عِيسَى وَفِي الْبَابِ عَنْ مُعَاوِيَةَ وَالْمِقْدَامِ بْنِ مَعْدِيكَرِبَ وَعَائِشَةَ . قَالَ أَبُو عِيسَى حَدِيثُ عَبْدِ اللَّهِ بْنِ زَيْدٍ أَصَحُّ شَيْءٍ فِي هَذَا الْبَابِ وَأَحْسَنُ وَبِهِ يَقُولُ الشَّافِعِيُّ وَأَحْمَدُ وَإِسْحَاقُ .
Abdullah bin Zaid narrated that:
"Allah's Messenger wiped over his head with his hands, going over the front with them and the rear. He began with the front of his head until they went to the nape of his neck. Then he brought them back again to the place where he began. Then he washed his feet." Abu Isa (Imam Tirmdhi) said: there are hadiths like this narrated from muawaiya, miqdaam, and aisha (ra). This hadith from Abdullah ibn Zayd is the most authentic on this topic, and Imam Shafi, Ahmed and Ishaq (ibn Rahwayh) are all of the opinion that the neck should be wiped in Wudhu.