You may be headed down the wrong CPS path here.
"no cranking" has NOTHING to do with the CPS.
The CPS only comes into play IF the starter is turning the engine over.
Basically, the CPS is used to tell the PTCM the crankshaft position so it can make timing and injection decisions.
The engine HAS to make at least 2 revolutions by the starter before the PTCM has all it's ducks in a row from the CPS signals.

You have what sounds like an RCM cracked solder or flaky relay problem, IMHO.