I haven't seen a trailer with a fuse in the brakes, blinkers, or backups. I've only seen them powered and fused by the TV. Then, I haven't seen all trailers.
I have seen TVs with and without a relay as KD4UPL said.
Still not sure of your statement "RV has running lights just none at the 7 way trailer plug". Sounds like you have a trailer plug "recepticle" on the back of your trailer to plug the carrier into.
If so and you have trailer running lights then it is a bad connection between the recepticle and the trailer lights. Trace that back to where it is connected to the trailer light wiring.
Since you said you wired it, I have to ask, did you use the carrier as a return path for the light grounds? I've seen that done, but the ground wire from the plug has to be grounded to the carrier frame also. I've always found it best to run a ground wire rather than use the frame.