Problem with the common inverter generators to power the house -- no 240v.
Even with gas heat, all of the several homes I'm familiar with still require 240v to run the gas furnace. It's stupid, but that's the way it is. When I replaced my system a few years ago using water to air heat (heat supplied by external system) I even asked them for a 120v air handler and it was written in the contract. But then they came back to renegotiate the contract because they couldn't get manufacturer warranty on a 120v setup and would have had to cobble the system together on their own.
Solve the furnace problem? Okay, next you get to deal with providing power to both sides of the panel. Normally being powered by 180d out of phase 120v, trying to power both sides with the same 120v appears at first to work. Just hope you don't have any circuits wired with a shared or down-sized neutral. If you do, you can now overload the neutral without triggering the over current protection for the circuit. Not a problem normally because the loads would balance and the neutral would only carry the current delta, but that isn't valid when the hot legs are in phase.
I do have a 120v to 240v transformer (4 of them, actually) to use with a 120v supply (in my case a large sine wave inverter) but the transformer wastes a lot of energy when idle. A not insignificant part of that loss is acoustic and that is intolerable. I do use a transformer when I need my well pump to run from 120v, but I have to manually switch it. I tried to rewire so the pressure switch would switch the 120v side. Even with the contacts in parallel they still burnt out dealing with the over 2x current demand. 50amp contactors are expensive, large and noisy.
Might as well get a proper 240v generator and solve all the problems at once.