seems there are more questions than answers.
first, a CAN bus lamp is pretty common these days. not sure if you can just swap out the lamps. but, if that is possible, then you certainly do want to use an Arduino for this.
as was mentioned. a simple remote, you can get a 4 button as they are pretty common.
left / right / brake/ running light...
1-2-4-6-8-CH-Channel-Wireless-RF-Remote-Control-Receiver-Relay-Switch-315-433MHz/
EBAY
https://www.ebay.com/itm/311768268377
if you have a jump box, that would have all the power you need and the Arduino could be connected to it with a simple cable and then have the output for a plug.
you could have a simple plug and then use whatever adapters, or you might be able to have all the adapters hard wired and then you just pick the one you need for that trailer.
Getting past the bit of if the lights are working, Wolframore offered to test resistances. this is a great idea, it will help with possible bad wires. you could put in it test mode, the wiggle with wires along the trailer and if there is a bad section the resistance would change and send an alert.