You don't need that converter. Plus the fact it won't work. It is solid state for 12 volt negative ground. If you put it in you will let all the smoke out of the wires. There used to be a Lucas service part to repair that problem but I haven't seen it for years.
More on Lucas problems: http://www.mez.co.uk/lucas.html
Wire your signals as I showed and it WILL work. You are setting up two completely different systems. One for the stop lights with the tail lights in them which you already have working and the other is the independent signal lights. This is why you don't use the brake light switch in the signal light circuit.
Your X638 switch is an all mechanical device. There are no solenoids, relays or diodes in there. It consists of an stationary insulated board with brass button head contacts in it connected to the four signal lights. The turn signal leaver moves a set of wipers or fingers which are connected to the flasher and when used to the stop light switch. In the neutral position the brake wipers are in contact with the rear signal contacts. When the brake switch is closed the signal lights serve as brake lights. When moved to the right turn position the left brake wiper remains on the left signal contact and the right brake wiper moves to a no contact position. The right flasher wiper moves to the right signal contact and the right signal states flashing. If the brakes are applied at the some time the left stop/signal light will remain on. The same idea works for the left turn. By not hooking the red wire to the brake light switch the circuit works the same but no power is on the brake wiper.
This should be as clear as mud.