I would start by checking your bulbs and sockets. It's possible that when you plugged into the harness it cooked both bulbs. Unlikely, but possible. If the bulbs check okay, use a multimeter to confirm that the socket ground is still... well, grounded. Just check for continuity between the outer wall of the bulb socket and any chassis ground.

Next, I would use your multimeter to check for voltage on the reverse light signal wire in the harness where you installed the trailer light plug. You'll have to turn the key on and place your AX15 in reverse to do this.

Next, check your reverse light switch wiring. The switch is located on the passenger side of the AX15 near the back of the housing, it will be the only wires coming out of the tranny. Make sure the wires are in tact and that the connector is in good shape and not full of mud or corrosion. If all that checks out, disconnect the switch from the harness and pull it out of the trans. You can test it with a volt meter, just probe both pins and make sure there is continuity when you depress the ball bearing at the working end of the switch.