A few things seem to jump out on first read. One is how the inverter was used as I would not recommend leaving it on full time as it does have the hazard of running the batteries down when things go wrong. Second small point is that it is using some power even when not needed if it is on full time. This produces heat and extra wear/tear on the inverter if it is not actually needed. Consider turning it off, much like a light bulb during the daylight, then turn it on when needed as a bit safer way to go?
But for really sorting an electrical problem, a meter is often needed as all the other ways to monitor do not get down to the specific points we need to test as they are located away from those points we need to test. We CAN do some looking without a meter but it does throw in more guessing than most like to do.
From what is described, it does seem to fit to look at the wiring at the transfer switch.
But a check of the main breakers at the load center is much quicker and easy as a way to avoid overworking!
At the load center we can get some breakers that do not quite act the way they should when they flip off or on, so it pays to check before working too hard.
Since you mention 50 amp cord and only plugged to 30 amp through the adapter, some things will not work but I assume that is something you are aware of, correct?
But if both legs are out, it is possible the breaker may need checked as the the two 50 amp breakers are tied together and should both flip off or on. But life is not always as it should be, so we need to try flipping these two main breakers (under one handle?) off and then make sure they both flip fully on again!
I suspect this may not be the problem but so easy compared to the next step!
If no joy there, taking the cover off the transfer switch may be needed! Disconnect the shore power cord and make sure the generator and inverter are off to kill power going to the switch. Safety first! Make sure the genrator doesn't com on while you are tightening screws!
The 50 amp RV should have something like this at the transfer switch with cover off.
Power should reach this point, blue from generator or green from cord, depending which is available. The transfer switch should connect the correct one on through to come out on two red marked lines going to load center breakers. Two legs of 110power to each of the two 50 amp breakers you found. Also there is the neutral in between on all of these points that act as the ground side. It's also very important to get good contact!
First good idea is to look for any signs of arcing like black spots at the points where wires connect. If finding none, it is often a good idea to do some preventive work while there. As we drive, RV get lots of shaking that can make things come loose and the screws where these wires clamp are one that do get loose over time, so I feel it worthwhile to tighten them while I'm there looking! No need to crush the wires, but just good and snug may put off the later trouble of loose connections? Don't miss snugging up the center which are neutral as well as the hot wires.
Another less common issue here can be simple failure of some of the contacts in the relays, either not making good connections or there are times when they stick and don't switch correctly. One to consider might be if the transfer has stuck on using generator power and you came back, shut off the generator and it did not work on shore power? If the contacts are stuck on generator power, you are still dead if you turn it off and plugged the cord in as it is wanting generator and not caring about shore power!
But this is where we may really need a meter as simple things like a broken wire in the cord plug can ruin our day and not being able to see that, we need to make sure power is getting to the tranfer switch!
Weird things like a broken wire at the campground plug can cause us all kinds of grief if we can't test and measure what is going on! A little like guessing at what our blood pressure is doing. Just in case you are the right group to know about that sort of problems?
