Darklock, something about your first post still just doesn't seem right and I wonder if you have more than one issue going on. Yes it really appears that the battery is a problem. But you also mentioned that sometimes all the 12v works after towing, and sometimes nothing 12v works after unplugging from shore power.
I'm wondering if you might have a failing circuit breaker (CB) between the battery and the trailer. Usually it's a small box on or under the tongue and on the positive cable from the battery.

If not the CB your built in charger might not be working.
Since you can take voltage readings here's a few checks that might help.
If you have a CB as pictured above, take a voltage reading from each post to the ground terminal of the battery. The voltages should be the same. If they are different then buy a new circuit breaker. CB's are available online as well as at places like O'reileys, Napa, Autozone for about $10. Just make sure you get the same size breaker as printed on the side of the old one.
With the shore power unplugged and the umbilical disconnected take a voltage reading at the battery (positive to negative posts) Then plug in the shore power and take a second voltage reading. If your charger is working and all the fuses and circuit breakers are good your voltage should jump up to at least 13.2v and hopefully higher. If no jump in voltage then you have a charging problem when plugged in.
Now disconnect shore power and plug in the Umbilical and start the tow vehicle. Does the trailer battery receive 13.2 or higher voltage?
I'm thinking that one of the above tests will show that something is not charging the battery like it should. Last thing. I would trace the negative cable from the battery to the frame. Remove that ground bolt, inspect for or clean off any rust or paint and then reinstall the bolt. This will help ensure that you have a good ground path back to the battery.