Yes, you need a converter. The VW has separate bulbs for the running lights, signal lights, and brake lights and the four wire trailer has running lights, but the signal and brake lights are combined.
You could use relays to make a circuit so that when the brakes are pressed, the brake lights come on, but when a signal is turned on, the corresponding brake light flashes off and on with the signal. You also need the signal lights to flash on and off when the brakes are not pressed.
Notice I say flash "off and on" when the brakes are also on, and "on and off" when the brakes are not on.
This is hurting my brain.