Chaosmagnet, I don't think the furnace will work on a 1000 watt inverter. I'm assuming that it will use 500 to 600 watts running, the starting watts could easily be 3x the running watts. Some inverters have fairly high surge capacity but I usually don't see that in a small inverter. Check it out before you rewire it.

Pigtails are a good way to go for controlling loads. I've permanently installed pigtails on certain appliances so that those loads can be run independently from a generator. It helps with load management, it's only plugged into the generator when you need to use it. I'm able to run everything in the house but the central AC (small window shaker for backup) and the deep well pump on a 3k generator. I back feed the fresh water rv tank for the house water, this works well for the short term.