If it was me, I'd start from scratch on the wiring and yes, a relay for each lamp.
Easy way out would be to have to turn the lamps on after ya get her started. Would go out when the key is turned off.
1. Turn key on.
2. Turn lamps off maually.
3. Start the beast
4. Turn the lamps on.
Crude, but it will work.
Otherwise y got to go into the headlamp (current) and find the wire that kills the lamps when ya hit the start.
My next question, is what are you going to do with all those wires in the stock headlamp? Hadn't thought of that had ya.