Where to start with adding a 12V battery to this system?

Start with heavy duty research on today's battery types and solar panel systems. There is a lot to learn and is fun to see how many differences there are.

Also look into your older converter. Converters charged a battery while on shore power. The battery often was used as a "buffer" for clean 12 volt power. There might be a place where a battery was, usually on the tongue. Lead acid batteries require a vented compartment, so are not the best inside the trailer.

By the pictures, a battery may never have been connected to your system. (12 volt output plugged). The outlet on the top was probably directly connected to the converter 110 power and only worked when on shore power. The switch appears to operate a transformer for 12 volt fixtures or power from the battery only. (trans/off/batt)

Check to see that the converter is putting out 12 volts. If so, all you need is a charge line to your new battery and a solid ground to the trailer frame. Some converters also have a ground directly to the frame. I think the bare copper wire going into the converter box may be a ground.