If the engine runs OK once it has started, then the FPR is serviceable to that extent. It may well have no fault, but we don't know. If the cold start valve works when +12V is applied to it, the problem is somewhere in the circuit between the relay and the cold start valve.
In the short term I'd try hard-wiring the valve direct to the +12v supply on the starter motor solenoid, as has been suggested; sounds like a very viable demon bodge (not botch...) to me. If that works, the only downside that I can see will be that the cold start valve will always add extra fuel, even on a hot start, but of course, will stop doing so as soon as the starter motor stops. I can't see a problem; does anyone have any other views?