Connection from the controller directly to the battery is always better. You will be giving up charge current because of voltage drop in the wiring unless the owner rewired the trailer to use thicker battery cables from the controller to the distribution and back to the battery. You will get some charging, it just won't be optimal.
If it was my trailer, installing bigger cables from the controller to the battery would be one of the first mods I'd do. With what you have for modules and batteries, that is going to be a nice boondocking system.
Yes, you would leave the batteries connected for charging while in storage. It is one of the best advantages of solar charging. You may also wish to add a switch between modules and controller (for battery maintenance or replacement, modules removed from circuit and then controller, it can damage controller if done any other way). You may also wish to add a fuse between controller and battery or 12v connection (if not present). Remember though, modules off line, first, and then disconnect controller to battery or 12V connection (remove a wire from modules).
Good luck with your new trailer and the solarsystem!
