Is it really reasonable for the downstairs neighbors to insist that I not use my shower because it leaks?

tell your neighbor it is unreasonable to expect you to not shower. if he prefers you not use yours then ask him to let you use his. if he refuses then use your shower. so far this guy has done nothing to help his own situation. if at this point he is still unwilling to do anything to help the situation then its his own fault if his floors get ruined.