[oil pan 4]

The battery problem is likely unrelated. When batteries get about 4 years old, they can quit with out warning. This is what happened to my last battery. It was working fine, luck for me it was a Saturday and didn't have to work, I came home from grocery shopping, then was going to go some where else an hour or 2 later, no start and the battery wouldn't take a charge.
The battery was starting the car perfectly before that, no warning what so ever it was getting weak.
Some batteries die a long slow death giving plenty of warning before their demise, other times they almost completely fail with no prior warning.

I bet they pulled the wiring harness till something broke trying to get some slack so they could cut into it.
Unless you are really good with vehicle electrical wiring it needs to go to the dealership.

The place and people who did your trailer wire install were likely highly rated because they kept all the most popular harness adaptors in stock, stuck them on and everything always worked as advertised. Then when they had to actually do something original, that required thinking and a little creativity such as custom wire in a P/T trailer hitch they were lost.

See if the bulb socket has continuity to ground.
This is the most likely scenario I can think of off the top of my head where the tail light wouldn't work but the trailer plug would test good. If I read your post correctly.

Also what kind of trailer connector did they install for you?
Is it a "flat 4"?