Thanks for the input so far, folks.
Any solution is somewhat complicated by this being a rental house - and there's definitely some dubious wiring in the loft; but that's a lighting circuit that has nothing to do with the fact that it's the ring-mains that get tripped, lights stay on. The landlord has been pretty responsive to issues so far, so I guess I'll fit the UPS in the office to protect The Money, and start sending emails to the landlord.
Kim's idea for protecting the Pi-of-Awesomeness (DNS, DHCP) with a power-pack was a life saver (thanks Kim!) as *nothing* works if that goes down; even if the router doesn't start after a trip, we still have local networking.
It's inconvenient as much as anything. I can't risk turning the power-shower on if fboab's on a Zwift race, and everything has to be turned off if we want to start the oven.