The trailer lights are wired in parallel with the towing vehicle tail lights.
If the trailer adds too must load (not likely) just replace the trailer bulbs with LED and replace the TS flasher with an LED flasher.
Those diagrams won't work because the TS switch in the towing vehicle controls the brake light circuit.
Typical problem with trailers is a poor ground. This causes all kinds of strange things. Make sure the trailer has a good ground to the towing vehicle.