Yes Another Generator Question

I know I know another generator question and also my first post. I have a 2006 Pioneer Fleetwood 180-CK travel trailer. We have had this for three years now and love it and 90% of the time we dry camp with it. I also have a 100 watt renogy portable solar panel that does the trick of keeping the batteries topped off before the night 90% of the time. Batteries are two Super Start group of 27 12 wired in parallel. I have only run the AC once when at full hookups and that was also the only time we have run the microwave. We won't ever run AC or microwave when not on full hookups. No TV, fridge runs on propane and we use lights occasionally (kids actually have AA battery LED lights they prefer as they are kids) so we don't even run the lights all that often. We have occasionally charged iPhone or iPad off of 12 volt plug in. We are low power people when camping.

However, I am considering getting a very small generator as I have had issues a few times at high elevations where when running the furnace at night and then a streak of cold cloudy (sometimes rainy) days (two or three in a row) where the solar panels don't get the chance top off the batteries to get through the night with the furnace running. Over these periods I have twice had to plug into and run the truck, which sucks as it doesn't charge well and just stinks overall. So basically all I am looking for is a small generator that would top off the batteries in the above described situation? Would a little 800 run watt generator work?

I know the honda 2000 would be great, but don't have that kind of cash right now. Would love something in the $250 range. Also, my other huge question is what what if any other equipment do I need exactly to make this happen? Can I just plug a solid extension cord in to the generator and then plug that into my 30amp trailer cord with the rubber yellow 30 amp to 120v converter I use when plugged in at home to gear up?

Thank you thank you for any advice or suggestions.