The OP should be Jumping on this. Inverter and 240v for under his price range.
See if you can figure out the circuit breaker capacity on the 240V side. Some have "ambitious" ratings that don't match the breaker capacity on the 240V side.
To some degree, what you are paying for with higher dollar lower power generators is "quiet". That doesn't seem to be an issue in this case. I'd personally, be looking for something around 10KW, if possible, I'd want as close as possible to 50A 240V output.
This assumes that you're going to connect it to the home with a generator interlock. If you just want a backup generator to power a few things (like a refrigerator / etc) - I think there are better options in the Champion family at power points around 3000-4500 watts. Much better fuel consumption. You can even get them in inverter format with dual fuel for under $700.
Power from non-inverter 3600 RPM generators is pretty "dirty" - I've never tried to fire home HVAC or ductless systems using it...
If you have a propane tank, definitely consider something that can fire on propane. 5 gallons of gas on a 3500 watt generator might last 8 hours or so..