I see a few potential problems here. First of all the little lights on your battery meter are pretty much worthless for anything other than a very vague generalization of the health of your battery. For anything remotely specific you need an actual volt meter.
You apparently have mismatched batteries hooked in parallel. If one is 5 years and you don't know the other one I must assume it is likely even older. They both likely are near or at the end of their life.
The outlets in most campers, particularly truck campers will not work on battery power. To do this would require an inverter, something just about no TC comes with. They are usually only factory installed in high end class A motorhomes.
All of your AC outlets and appliances should work from your generator. I would think that most campers would have an automatic transfer switch. Yours may not be working. Alternatively there may be an outlet in a compartment near the generator that you have to plug your shore power cord into to get generator power. This is a common arrangement on Class C motorhomes. Assuming you have an automatic transfer switch it likely has a bit of a delay before connecting to the generator, possibly as much as 30 seconds. You might not be waiting long enough.
