Permanently installed appliance powered by a portable generator? I would not expect anything different when the power is out due to a storm.
Heck, I spent Dec. 26th without power for eight hours, in 18F weather. If I had a portable generator available to power the fan and controls of my gas furnace, you bet I would have passed an extension cord through the window, in a heartbeat.
Indeed, I'd prefer furnaces to not be hard-wired, to be plugged into a standard receptacle, for this reason. I don't want folks tying the generator into the main panel with jumper cables or suicide cords.
So, I'm rather distressed to hear that a quality generator won't work- but a cheap one will. That's quite a design issue, IMO.