Is it permissible, in Islamic law, to undergo medical examinations to determine the cause of feeling a complete inclination to transition to a man? And if permissible, whom in Cairo can one consult to perform these analyses?

There is no harm in undergoing medical examinations and analyses by specialist doctors to determine the gender. One can treat the soul with patience, by resorting to God, by keeping company with the righteous, and by seeking refuge in God from Satan and his whispers.