Ok, good feedback & good to have tested the fuel pressure with a gauge at the fuel rail. 53-63 psi is the range, so something is amiss with the pump operating voltage, or the internal fuel pressure regulator.

Measure the alternator output voltage under load & measure ripple, in case you have ripple / noise from faulty, or leaking diodes, that's causing problems on the B+ that's carrying over into the ECM, or anti-theft system. Anything over half a volt ripple riding on the alternator B+ output will likely cause mischief.

Do you have another known good ignition key that you could try, in case the one your using has lost its code & is triggering the anti theft to kill the engine???

More thoughts for consideration, let us know how it goes.

If all continues on the current path, this will be the final post on this issue. I have been driving the Neon now for a couple weeks after the deletion of the SKIM and the install of the remanufactured PCM and zero issues of stalling, no cranking or no start. Also replaced the fuel pressure regulator and restored the restored to fuel pressure to 53-60 psi, short term fuel trims are in low single digit (+) (-) and long term is slowly returning to the normal. I have received a catalyst infencey code but that was on it's first long 30+ mile non stop drive. Cleared and have driven a couple long trips and no codes returned. Thinking it was maybe just burning off excess fuel. If not we may be looking at a converter in the future.
Paw Paw thank you for your input. hope this post may help someone in the future. Soon I will pull the instrument cluster and see if the security lamp can be removed to rid it of that pesky red light. Some post say it is a soldered in LED some say a PC194.

OK, good trouble shooting feedback & to hear all seems to be going well. If the problem returns, come on back here & we'll think about it some more. Good feedback & idea to have put the fuel pressure anomaly right. Sounds like you may have put the fix on this puppy.