The hadith does not prove that the bedouin wanted to renounce Islam. Rather from the context it only shows that he wanted to leave Medinah because its climate did not suit him. The Bay'ah (pledge) included the promise of remaining in Medinah as migration was obligatory at that time. The bedouin's action of leaving Medinah was fisq (disobedience) not kufr and hence he was not an apostate. Hence this hadith does not change the ruling of punishment on apostasy.
Secondly if for argument's sake we assume that the bedouin wanted to leave Islam: the hadith will still not be acted upon, because:
the punishment of apostasy is proven through tawatur (multiplicity of sources) and ijma' (consensus), one khabar ahad (solitary report) is not sufficient to overrule it as there is greater probability for a single person to make a mistaken in the transmission or comprehension of the report.
it would be a contradiction between the prophet's command to us and the prophet's own action in a specific case. In case of such a contradiction we are to follow the command. For example if the Prophet told us to not marry more than 4 women, but himself married 9 women, then we follow the command.
it would be considered abrogated: since the later actions of the Prophet at the conquest of Makkah, his instruction to Muadh bin Jabal when dispatching him to Yemen as governor, and the actions of the sahaba during the Ridda wars proves that they punished apostates.
For further references on apostasy you can see: Is punishment for leaving Islam death?
The Pledge of Islam included other promises apart from belief in Islam, such as the promise to refrain from theft, fornication, murder, slander and other disobedience (60:12). Similarly it included the promise to participate in Jihad and at times also to migrate and stay in Medinah (see e.g. Bukhari 4305) and this was the part of the pledge taken by the Bedouin which he wished to cancel:
وقول الأعرابي: (أقلني بيعتي) يريد: أقلني ما بايعتك عليه من البقاء بالمدينة
The saying of the Bedouin (Cancel my Pledge) means: cancel the pledge in which I promised to you that I would stay in Medinah.
—Mu‘lim bi Fawaa’id Muslim
This is also apparent in a wording of the hadith in Musnad Ahmad and Musnad Humaidi:
جاء إلى رسول الله صلى الله عليه وسلم رجل من الأعراب فأسلم، فبايعه على الهجرة، فلم يلبث أن حم فجاء إلى النبي صلى الله عليه وسلم، فقال: أقلني، فقال: " لا أقيلك "، ثم أتاه، فقال: أقلني، فقال: " لا أقيلك "، ثم أتاه، فقال: أقلني، فقال: " لا "، ففر، فقال: المدينة كالكير، تنفي خبثها، وتنصع طيبها
A man from the Bedouins came to the Messenger of Allah ﷺ, accepted Islam and gave the pledge to emigrate. Then he developed a fever so he came to the Prophet ﷺ and said: "Cancel my pledge". The Prophet replies "No". This happened three times and then the man fled from the city. The Prophet ﷺ said: "Medina is like a furnace, it eliminates its impurities and purifies what is good."
— Musnad Ahmad
قدم أعرابي المدينة، فبايع النبي صلى الله عليه وسلم على الهجرة، ثم حم، فأتى النبي صلى الله عليه وسلم، فقال: يا رسول الله أقلني بيعتي، قال: «لا» ، فلما اشتدت به الحمى أتى النبي صلى الله عليه وسلم، فقال: يا رسول الله أقلني بيعتي، قال: «لا» ، ثم اشتدت به الحمى، فأتى النبي صلى الله عليه وسلم، فقال: يا رسول الله أقلني بيعتي، قال: «لا» ، ثم اشتدت به الحمى، فخرج هاربا من المدينة، فقال النبي صلى الله عليه وسلم: «المدينة كالكير تنفي خبثها، وتنصع طيبها
A Bedouin came to Medinah and gave the pledge to the Prophet ﷺ to emigrate. Then he developed a fever so he said: "O Messenger of Allah Cancel my pledge". The Prophet relpied "No" and his fever increased even more. This happened three times and then the man fled Medinah. The Prophet ﷺ said: "Medina is like a furnace, it eliminates its impurities and purifies what is good."