It sounds like you have a short in the wiring going to the right turn signal/brake light. I would confirm that you have a short by checking with an Ohm Meter. Check for resistance between the right turn signal feed and ground feed at the plug on the trailer. If it is zero, you have a short somewhere. It might even be in the bulb, so I would check there first. I haven't seen but one or two, but is is an easy place to start. If that isn't it, remove the tailight and see if something is pinched there. If you haven't found it by then, you might try a short finder. I don't know if they will work thru the interior aluminum, but it is worth a try. They can be purchased at J.C.Whitney or from a automotive tool supplier. The short finder sends a power pulse down the wire. By using a sensitive inductive meter (part of the kit) you can follow the wire with the short until the meter no longer deflects. That is the location of the short. Of course, the problem then is getting to it!! I had an intermintent short in the tailight circuit on an older Coleman that I had. It wouldn't stay shorted long enough to find it, but would consistantly blow fuses going down the road. I finally ran a new wire from front to back and isolated the old wire to fix the problem. You might find that you will have to do the same thing.
Good Luck. You will need it. Finding a short is always difficult, even for the experienced.