The way that you have shown the wiring, all three trailer lights are in parallel and will light at the same time.
If you have a diode in each of the two positive wires, then the trailer lights won't be affected by the additional light, but the central LED light will flash when the brakes are not on and the car is indicating.
The tail light converter should allow the central LED light to work as a brake light only, but the dimming might still happen. That is due to the voltage drop in the wires, caused by the resistance of the wires and the current taken. To stop that, either increase the wire size, or get an LED light that has better current control, so that its light output doesn't vary if the voltage varies.
You could also use a connector that is separate from the standard towing connector to join the LED light directly to the vehicle's central brake light.