Update: I got the replacement wiring harness and was having the same problems with it.
I did some more troubleshooting and made some progress - the trailer has 2 brake/turn lights on each side (4 total). If I took out a bulb on each side everything would work.
I found on the CURT website it says the brake and turn circuits are 3 amps in the wiring harness. Each of the 1157 bulbs is drawing around 2 amps a piece, so with all the bulbs in the wiring harness was overloading the wiring harness and shutting the circuit down.
I'm going to try buying LED replacement bulbs for the trailer and see if that will work.