Pilot steam heat backup AC source(UPS) to run thermostat and transformer?

Hi, I have a vacation home that is heated with gas non electronic pilot steam heat (burns a pilot 24/7) so no forced air( no digital thermostat just the old round honeywell style with a voltage transformer on the input side ) .

I'm wondering would it be possible to run an inline APC/ UPS type unit (similar to what's used to run a computer during a short power interuption) to simply keep the electronics running (which in this case is just the thermostat and anything forward of that from the electrical panel to the furnace. Plug in the APC to a nearby outlet then connect the 110 out put side of the APC via a "cheater style jumper from the APC's ac outlet into the input side of the heating system?

So currently if power were to fail I'm guessing now the furnace wouldn't turn on because there's no power to the thermostat side, but the pilot is still just sitting there burning , waiting for input from the thermostat which is dead due to the electricity being cut. But if an APC/UPS were to be connected running day in day out like you would for a computer , would this work for a short period (depending on amp draw of the control side and battery capacity of the APC/UPS) to operate that boiler's control side? The house may be out of power but the house would still have heat via the power supplied by the APC.
I'm guessing most APC/UPS are modified sine wave not true sine wave , are thermostats and the transformer parts able to be operated on a modified sinewave power source as a maintenence power back up or thru put from the wall into the APC/UPS then to the control side of a heating system. (again NON forced air steam heat).

Thanks for your time