I spent some time thinking along these lines, but the scale of the problem made me cautious -- with over 100 entries per event, there are a massive number of possible combinations. Even with a fair amount of optimization, we're probably talking about days, or at least hours, of computation to work through them...

And given the real-world possibility that some teams may be added at the last minute or fail to show up on the day of the event, you'd probably need to pre-calculate the choices for different numbers of entries from different numbers of schools, which means repeating the above computation thousands or millions of times.

Given those constraints, it seemed the best real-world solution for the OP would be an approximate, non-exhaustive one -- although I'm sure that the solution I posted above could be improved with further effort.