Why is everyone overthinking things? If all you want to do is keep the batteries charged and run the fridge on LPG (control board will use a small amount of 12v), run your 100' extension cord and plug your trailer into it using a 15A/30A adapter (available for about $5), and let the trailer's charger/converter do it's job. You won't be able to run microwave or Air Conditioner, but for what you want, it will work fine.
It's exactly what I have been doing for 28 years and 3 RVs during the week or 2 leading up to a trip, and even running the RV fridge on electric to pre-chill it.
A typical RV fridge uses less than 3 amps on 120VAC; and a 45A converter will only draw about 5A at peak charging.