Who would have guessed?
The dealership never opened up the junction box on the trailer. If they had, they'd been able to figure out the issue.
I spend 9 hours fighting with the wiring today, but made progress tonight.
Had to hook the green right turn wire from the 7-way to the brown trailer wire.
The yellow left turn wire from the 7-way to the red trailer wire.
The brown wire from the 7-way ended up being the brake pedal wire, so that needed to be connected to the (2) green trailer wires.
The red wire from the 7-way is the aux power that's in sync with the cigarette lighter, so that connects to the thick black wires that connect the trailer battery to the trailer and also charges the trailer battery when live.
No wonder like colored wires connected together caused the issues!
But I don't have a wire that powers up when I turn on the headlights. There is a black wire coming from the 7-way but nothing sends power to it. For all I know at this point that black wire is the ground and the white wire is the headlamp feed. Out of daylight so I'll check that tomorrow.