Well, the brake circuit and turn signal circuit are one and the same on the trailer. Does the right side brake light come on at all when applying brakes? Or does the fuse blow immediately? If it blows immediately, I'd say the problem is in the TV. If it comes on for a short period of time and then blows the fuse, then there's something drawing too much current on the trailer. Although it doesn't appear to be a dead short.
By the way, you can "typically" just use the emergency flashers for a brake light/turn signal check since the flashers/turn/brake are all the same circuit/light bulb filament for each side. Although your problem is pretty strange that the bulb will blink, but blows the fuse with brake application.
03 Chevrolet Silverado 1500, 5.3 V8, 3.73
06 Rockwood Roo 23SS