Sounds like it's time to put a regular 7-pin RV plug on that trailer. The 7-pin SAE plug is typically on commercial plugs.
Note that the SAE version doesn't have all the same pins, besides the pins being round. Brake and turn circuits are separate (not a big deal... just reuse the brake light circuit as the trailer brake circuit, run the turn signals as combination stop/tail lights swapping to red lenses if it already doesn't have them), there's no backup-light circuit, and there's 2 running light circuits one for tail-lights one for clearance lights.