The little switch/red light is your TV antenna booster. It has nothing to do with the 12V plug.
If you were to take the entire plate off the wall, you'd find that it's a circuit board behind there. So if the power goes to the booster but not the 12V plug, unless it's an obvious break somewhere, it could be anything with the circuit board or elsewhere.
I think Paul's method is the best way to test it.
Just to make sure, do you have a battery on the front of the trailer? It may not work with just the converter 12V output (and no battery) because it may not be sufficient enough. I don't know if it really matters (never tried it before), but it may be worth a shot if it still won't work for you.
2007 T-286SR Cherry/Granola, #6236, original owner, current mileage: 9473.8 (as of 6/18/21)
1997 T-2653 Blue Denim, #5471
1979 12 1/2' MC, Beige & Avocado, #4639
Past Sunlines: '97 T-2653 #5089, '94 T-2251, '86 T-1550, '94 T-2363, '98 T-270SR