Lets all sit at our computers and guess.
NO.
Lets go thru this LOGICALLY.

1) The switch on the console locks and unlocks everything fine. This means: The console switch group, BCM and CLP/SSM work well together.
2) The key switch in the door does not work - this is wired directly to the SKREEM, which then tells the BCM and CLP/SSM what to do, but the SKREEM is not doing this.
3) The FOB is proven to be transmitting, so it's batteries are good and are contacting the tabs inside. While the key FOB might still be sending gibberish due to a bad board or corrosion, I doubt that.
4) It seems like you can't sync the FOB with the SKREEM, this is the second hint the SKREEM is at fault.

You either have a bad FOB (and you and I both doubt that) or a goofy SKREEM. Remember, there are two halves of the SKREEM, the RKE (Remote Key Entry) half and the "Secure Start" half that starts the engine and is the usual cause of the "Three cranks then nothing" start failure. In your case, it seems like the RKE half is insane.

I'd call Josh at the MSS and see what is diagnosis is.

You can yank the ECU/SKREEM/Ignition switch and key FOB from another car but even that has "gotcha's" you really don't want to deal with. I'd say it's easier to get Josh's help and probably no more expensive, as well as faster.