If there is a battery-disconnect switch, it was installed by a previous owner rather than the factory, so it could be anywhere around the battery or the univolt.
More likely is that there's a bad connection or open fuse between the battery and the main bus of the fuse block. One quick check for that would be to measure the voltage at the battery when the trailer is unplugged and then measure again when it's plugged in. If the voltage is about the same, the battery isn't properly connected to the 12v
system (the Univolt that's powering your lights and other 12v
stuff when plugged in should also be applying a charge to the battery.)
No one complains about 3-stage Iota or Progressive Dynamics converters. (Note that some of the cheap Progressive Dynamics models you'll find on eBay and Amazon are single-stage, pay attention to the model numbers.) I have had good service from a Powermax Boondocker converter, but I've only been using it a year or so.
If you have the original Univolt, the upgrade is well worth it both for the better converter/charger and for a modern fuse block that uses the better automotive-type blade fuses.