Not before you answer my post #22. I am still not thinking this is a SKREEM issue, as you said, the symptoms do not line up with a failed SKREEM. The system was designed to let you know if the SKREEM is inhibiting a start - the ECU is programmed to do the 'three cranks and nothing" if the SKREEEM fails to authorize a start.

You have NO STARTER ENGAGEMENT at al, from what I read. A failed SKREEM won't do that and I don't give a damn what the dealer says. (Unless there is some undocumented failure mode of the SKREEM/ECU interface that we have never seen here in the eleven years I've been here - what are the chances of that?)

Go to post 22 and do the test I outlined. If pressing the relay does not cause the starter to engage, we have what will be a rather easy thing to fix - your starter or solenoid is bad (very rare but who knows?)
If the starter DOES engage, it will be interesting to see if the engine also starts - if it does, we have a fundamentally simple failure between the ignition switch and ECU/Pulse module.

From post 22:


Now, you can take the Pulse module out of the fuse box, pry the cover off, turn key to 'on', and manually close the relay (there are two, one for the SRT circulation pump, one for the starter). If closing this relay makes the starter engage, you know:
A) The starter/etc. is OK.
B) The issue is somewhere in the intricacies of the electronics between the starter contact on the ignition switch/the ECM/the Pulse module/etc.