Extension cord wire gauge has nothing to do with tripping the breaker. It could affect the voltage available to appliances in the RV and may contribute to a warmer than normal charger/converter under heavy load due to the AC voltage drop in the cord but unless you have bad batteries that pull a lot of current from the charger, this is unlikely.
I believe you have other stuff using this circuit. 30 minutes of operation implies a over current load on that circuit. When the breaker trips, what else goes out? Over current loads are not going to be fixed by a heavier extension cord.
As you also guessed, a difference between the neutral and ground between the house and the RV might be the culprit. This normally manifests itself by the GFCI tripping in an outlet unless it is plugged into a circuit breaker that is the GFCI for the whole circuit. You did note that the GFCI doesn't trip. Noting this, I suspect one of two problems: Your circuit breaker is "soft" or trips with less load than it is labelled OR you are, in fact, pulling more current on the TOTAL circuit. What else is on the same circuit as the RV? Circuit breakers will deliver more current than their rated limits for a short period of time. If this wasn't so you would have to have an air conditioner breaker in the hundreds of Amps instead of 20-30 Amps. The A/C pulls a huge amount of current for a very short time when the compressor motor starts and then it settles down. Fridges and freezers and pumps all do this also.
From a safety side: The extension cord wire size should match the breaker current limit. Use 10-12 gauge for a 20 Amp circuit, a 12-14 gauge cord should be OK for a 15 Amp breaker. You don't want to fry your extension cord although being outside a wall and exposed to the air will help in cooling the wire under load.
Big current users in a RV are the A/C, Microwave Oven, space heaters, and huge TVs. A 50 Amp battery charger (big in the RV world) should be pulling less than 10 Amps, more likely less than 5 Amps.
A clamp on AC Current meter on the circuit in question will answer your question here. It is best done at the breaker panel. Something like this:
https://www.harborfreight.com/digita...ter-96308.html
You will be working in an area that can kill you from electrical hazards so if you are not comfortable with this - do not do this. This meter must go around only one conductor (black) of the circuit being examined. You can't snap it on the extension cord as bot the hot and neutral wires are in the same cord.
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