Bear (bare?) in mind too, other loads that might occur. Such as the converter that charges the batteries and runs most of the lights in the trailer. If the batteries are charging, or you are running lights,that adds a load to the 120 volt side. Also, the ceiling fan, a couple of the other lights, and other 120 volt loads come into play, IF someone forgets and trys to start them. With your 2000 watt generator, it should be fine for the "fireplace" and the converter. Just won't be able to turn anything else on.
Remember the old "Green Acres" show with Eddie Albert and Ava Gabor?
Remember that generator out back, and how they had to manipulate what they could plug in and run in the house???
Also, remember that your fuel consumption on the generator will go up considerably. Generator fuel consumption figures are usually based on only a partial load.