The first thing I would do is check the CG voltage with a voltmeter and also check for mis-wired pedestal. Inside your TT, like at a recept., check the voltage hot to ground, hot to neutral and neutral to ground to see if anything is unusual.
Lights going out because the AC supply turned off does not make sense. The lighting is an independent 12 volt system. If the microwave comes on okay, then the AC system is likely okay. Doesn't make sense that the toaster oven would trip the 30 amp main brkr.
Does anything besides lights that requires 12VDC, like the fridge or furnace work? Won't be the cause, but check the DC fuses in the panel. If you disconnect the positive at the battery, do 12 volt things work okay? If you have the battery connected and shore power turned off, does the 12V system work okay? A dead or weak battery would not be the cause of the breaker tripping, but could be part of the answer to the puzzle. With the battery disconnected, what voltage is the converter putting out? I would check and clean connections at the battery and to the frame. How old is the battery?
*If* an AC only problem: Start by turning off ALL branch circuit breakers in the panel in your trailer and leave the 30 amp main ON. If it's stays on, turn on a branch breaker one by one until a breaker trips (if it does). If there is an overload or short on a branch circuit, it should trip the branch circuit breaker only so if one does trip, that's the circuit with the problem. If there is a constant buzzing sound coming from the panel when you have more than one branch breaker causing a buzzing there is a breaker problem. If a breaker buzzes and then quickly turns off, then you have a problem in the circuit itself.
The only thing in common between the AC and DC system would be the grounding at the ground bus located behind the converter panel. There could be a loose ground and you can check the tightness at each terminal. I would also check the ground connection from the bus to the chassis/frame which would a lug connected to the frame somewhere near the converter panel.
If you are at a pedestal with two 30 amp receptacles in it, try plugging into the other one.