It sounds like you may be new to campers as well?
The 12volt Side of Life (Part 1)
I like recommending that page as a starting point for familiarizing yourself with the 12V DC power system in a typical camper. Without knowing how your system is installed, it is impossible to make definitive statements. But hopefully reading through that site will get you familiar with the basics.
Don't forget to proceed to part 2 which touches on inverters and solar.
The 12volt Side of Life Part 2
...We can see on the Zamp panel that we have a full solar charge of 13.5V but can't seem to figure out how to switch over to the solar-powered batteries. Do these batteries run the A/C and outlets? We have other questions but this one is the major one at the moment...
I think the info in the links may arm you with the knowledge to answer these questions, but there are so many "it depends" caveats that can't be answered without knowing how your system is installed. I also don't actually know what the Zamp kit is you have mentioned, but I assume it is solar panels, separate Renogy inverter, and the 3 batteries are installed on the camper? Normally you have one set of 12V batteries (or 6 volt batteries wired in such a manner to provide 12V), so there wouldn't be a situation of "switching over to the solar-powered batteries." There would just be your batteries, which provide the 12V DC power to your camper. The 12V batteries will need recharging which is done by your converter when plugged into shore power/generator, and in your case should also be getting charge anytime the sun is shining via your solar panels. The inverter can take the 12V DC power and invert it to 120V AC power for your use. Now how your inverter is installed is still a mystery at this point, pictures would help. But generally an inverter would NOT run the air conditioner, that would be too big of a load for a battery/inverter system. If your inverter powers the AC outlets is also a mystery, it depends on how it is installed. It sounds after market, so really it was what the previous owner felt was appropriate.
The 13.5V is the voltage of your batteries. That value does suggest that the panels are providing charge to the batteries if it is being read when you camper is not getting power from the generator and it is during daylight hours. But it is not a full picture of your battery state of charge (SOC) without knowing more. 13.5V may suggest full batteries in a float charging state, or it could mean batteries are in the bulk phase of charging and that reading will continue to rise.
Charging phases of a typical smart charger