Do you have a inverter in the RV?
Do your lights work when not plugged in?
There may be a 12 volt fuse between the battery and 12 volt fuse panel that is bad, if the lights do not work when unplugged.
This is my guess because I think what is happening is the 12 volt output from the converter is powering the lights, and not the battery, so when the converter is overloaded, it will stop producing power, the thermostat has no power to keep the A/C running, and the lights go off at the same time you are losing 12 volt power to the thermostat.
You might also have a shorted wire between the converter and 12 volt panel or the (more likely) 12 volt panel and the battery where it's fuse is located. Or you might have hooked up the new battery backwards, and when the automatic reset circuit breaker is tripped, you have 12 volt power to everything, but when it re-sets, you have no power as the converter is overloaded.
Just a guess there.
Good luck,
Fred.
Also if you have a inverter, shut it off. It might be what is tripping the circuit breaker?
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