I think the vendor's "burn out" warning was due to the higher amperage breaker on the 30A circuit. Since most 120V devices are designed for 15A and 20A circuit protection, you can easily overload components in your electrical system before the 30A breaker will trip. An example would be if you were running a couple space heaters and a microwave on the same extension cord. Normally, that configuration would cause the breaker to trip since you'd be pulling more amperage than the circuit could safely deliver. With the breaker upped to 30A, you could easily set your extension cord on fire with the added current draw before the breaker would step in. In short, be sure to carefully monitor your current draw. Add up everything that you have plugged in, and make sure you are not exceeding the weakest link in your system. If you are using a 14GA extension cord, do not exceed 15A total draw (1800 watt total). A 12GA extension cord will get you up to 20A (2400 watt total). If you have long extension cords, then the amount of power you can safely draw will drop. It doesn't sound like you're using any high-draw appliances, or are anywhere near capacity, but please be careful.
The issue with the charger could be bad polarity, as others have mentioned. It could also be low voltage on the line. As other also mentioned, campgrounds have notoriously bad power. Oftentimes, they don't use appropriate wiring, or over-allocate electric circuits. This can lead to circuits that are under-voltage. You can test the circuit with an AC voltage tester, or multimeter. A nicer option is to buy a
Kill-A-Watt ($20) electric meter. This will give you all kinds of nice info about the power you're drawing, including voltage, actual live amperage draw, frequency, etc.. It can also track total watt usage, which would be super-useful to help with sizing a battery/inverter electric system to play off-the-grid.