The cold start valve really doensn't do much unless you live in a very cold climate. IMHO, I suspect that you you are losing residual fuel pressure while the car is at rest via leaky injectors, or you have a bad fuel accumaltor("Accumulator- Large spring loaded diaphragm to keep fuel pressure up with engine off, also dampens pulses from the pump"). I would put a set of guages on the car first and see what happens.

Let us know what happens.