Trailer Light issue
I would also test the hazard lights with parking light on/off and see what you get.
However, since you only have a problem with one device, (right brake light) I'd suspect a wiring problem with that device or the wiring to that device. That said, because grounds are almost always the problem here is a good place to start:
*Check the ground wire (white) leaving the flat four. It leaves the flat four and connects directly to the frame of the trailer. Many times this connection gets corroded, or the pin itself breaks off in the female of the flat four. This connection is molded within the rubber - so it is easy to miss if it breaks off. Had me scratching my head for 30 minutes when it happened to me... Definitely something to look at.
*Next you need to check the right brake wiring. Go to Napa and buy a DC Voltage light. They are pretty cheap. They don't give you voltage, just light up if voltage is present. On the flat four leaving the tow vehicle the third female pin is green (right turn and brake), put the black lead of your testing light into the only male pin which is white (ground), and put the red lead into the third female pin which is green - then have somebody step on the brake of the tow vehicle and see if the light turns on. If it does you have eliminated the tow vehicle and it is time to move onto the trailer.
*Go to rear right side of the trailer and look at the wiring, make sure a connection didn't come loose. If all looks well, disconnect the wiring at the tail light. Put the red lead of the voltage light on the green wire, and the black end on the ground of the trailer, then have somebody step on the brakes and see if the test light turns on. If it does, you have a bad tail light, if it doesn't you have a wiring problem between the flat 4 and the tail light.
Check grounds, check fuses, check connections. Most of the time it's one of those. Also check your trailer with a different vehicle and check your vehicle on a different trailer. Very quick and simple and will isolate the issue as either the trailer or the vehicle.