For power you could grab the 12V charge line at the hitch, and you could probably tap into the backup wire at the hitch as well.
https://www.etrailer.com/faq-wiring.aspx
Two 18W lights (36W total) only consumes 3A. Wouldn't need a very big relay (or one at all, perhaps). If you swapped out the backup-lights with LEDs I'd bet the total current for the all-LED setup would be less than the original incandescent backup lights!
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