I haven't had this problem, but my suggestion would be to simply call the one you hope to change, and simply tell them the truth (you don't have to tell them which school the conflict is with, just state that you have another interview invite on the same day). Then, either they will tell you they can reschedule you, or they will say no, in which case then you could try rescheduling the other school before you have to just make a decision to tell one of them no.
Good luck - it's great that you are getting so many interviews.