Is it permissible to set a specific time for performing supererogatory prayers daily – such as twenty minutes – with the aim of maintaining them and avoiding fatigue, while not believing in its Sunnah, and is this considered an innovation (bid‘ah)?

There is no harm in performing supererogatory prayers between the two 'Asrs (Dhuhr and Asr), but persisting in it may lead to it being considered an innovation (bid'ah). Therefore, one should not consistently dedicate this time to a fixed supererogatory prayer. Rather, one should strive to increase supererogatory acts in general, without designating a specific time for them.