If you don't plan on moving it and have access to shore power, any 12v battery would serve since you will be getting 12 v power from the converter charger.
Since you've gone to the expense of replacing all the bulbs with LEDs, I assume you are using it "off grid" and need 12v power between generator, solar, or wind charging. There are a number of things to consider. You might check out a couple of 6 v golf cart batteries for more amp capacity. I'm sure many will tout their favorite battery brand, check on line for reviews of various batteries suggested and make your own decision. If you have limited space, you'll be forced to buy battery(ies) that will fit. If your truly not moving the trailer, then you could expand the battery storage and put in any size. You should also check out what kind of charger your trailer has, it might also need changing if you go to deep cycle batteries.
Here's a pretty good source of information an battery terms and factors to consider:
Deep Cycle Battery FAQ