Could I use a UPS to power my humidifier on an underrated breaker with a little trickery
No.

You definitely want to put the humidifier on its own circuit, or a circuit that is not shared with two powder/bath rooms. You could look for a convenience receptacle in the basement and determine if it's on the same circuit. If not, you can probably come out of that for the humidifier. If the humidifier pulls it's full 7A rating when it starts, you probably want to avoid having it share a lighting circuit, as the lights could dim when it turns on. That can cause premature wear on ballasts and greatly reduce the life of any incandescent lamps you might have in use. Take care to know exactly what else is on any circuit you find in the neighborhood.

NEC requires wiring devices, such as the relays you're talking about to be listed and approved. Assembling something to cut the power to the UPS when the humidifier kicks in using all listed and approved parts does not constitute a listed and approved device. Your approach, while very creative gets a little too far into the "Rube Goldberg" brand of tinkering. 

My advice is, have an electrician install a single use receptacle for the humidifier on its own 15A circuit, even if that's slightly overkill. If your panel will not accommodate this, and tandem breakers aren't a possibility, then consider increasing the size of your panel and / or service.

Two bathrooms on a single 15A circuit is problematic enough, it sounds like you need to bite the bullet and get a few additional circuits where they're needed.