Is it obligatory to re-pray Fajr if its time is about to expire, and purification from urinary incontinence took a long time, causing the worshiper to doubt whether he caught the prayer within its time or not?

If urine typically stops after ten minutes, one must wait for it to cease. The ruling for a person with urinary incontinence (sahib al-salas) only applies if waiting for it to stop would cause the prayer time to expire. In that case, one should perform ablution and pray. Based on this, what was done was correct if it was known that waiting for the urine to stop would lead to the expiration of the prayer time. Regarding doubt about the time still being within its period, the default is that it remains until its expiration is certain. Therefore, the prayer is valid and was performed within its time.