Your basic problem is that there isn't a brake light line on the trailer connector; there's just a left brake/turn and a right brake/turn. It sounds as though you just shorted the two together and fed that to your new brake light, and apparently the same circuits power the vehicle's turn signal lights. Even with steering diodes to prevent back-feeding, you'll still have the problem of the brake light on the bike rack blinking whenever a turn signal is on and the brakes are not on, say when changing lanes or merging onto the highway, which I suspect could be rather confusing to drivers behind you.
If you have a third brake light on your vehicle (or no third brake light but separate signal and brake circuits at each side), probably the simplest approach would be to tap into that line and run a separate connector for the light on your bike rack, not plugging into the trailer connector at all. If you don't have a separate independent brake light circuit anywhere at the back of the vehicle, it will be pretty hard to do what you would like to do; you'd probably have to run a wire to the front and tie into the wiring harness somewhere in the dash area, perhaps at the brake pedal switch, and the details on how that would work probably vary somewhat from vehicle to vehicle.