Is repeating the Asr prayer after changing the intention from obligatory to supererogatory during the prayer due to a part of the back becoming uncovered, and then doubting the validity of the second prayer, considered valid or void? And what should be done in such cases in the future?

The covering of what is between the navel and the knees, and what corresponds to it from the back, is obligatory for the validity of a man's prayer. If a small part of the back is uncovered unintentionally, and it is covered immediately, then the prayer is valid and not nullified. You have done well by repeating the prayer with the intention of fulfilling the obligation. You should turn away from obsessive thoughts and pay them no mind.