aha
if it is the cold start, that is an independant system.
I doubt a relay has anything to do with it, tbh.
But just for info:
your 03302014125 has these pins: 87, 87, 86, 85,30
any relay with that pin out will work in its place, no need for the bracket for testing purposes.
When the car is cold, remove the CSI and have someone crank the starter whilst you aim the csi into a jar......does it spray nicely?
if not, test the TTS this supplies the ground signal to csi