My bet is it would work fine off the generator, I do think that if you throw in the heater and lights it may get close on what the generator can supply. Especially if you add a/c to it at some time. I don't think you need to worry about extra equipment to catch any rogue voltage spikes.
The battery will act as a good buffer for any small voltage spikes. It's the large spikes you have to worry with. The type caused when some one decides they want to test their alternator by removing one of the batter cables from the batter to see if the engine will still run.
