Unknown fault codes

C1406-12 Transfer case brake control - circuit short to battery.
C140a-13 Transfer case motor control - circuit open.
C2110-12 Sensor supply one voltage - circuit short to battery.

All related to the transfer case and its controls. I doubt they involve the button but the last one that shows the five volt reference shorted to voltage could.

From the descriptions I would start by looking for harness damage near the T-case or its controller. A bad wire shorted to the others could be the issue.

Just gave you the information so you knew what the codes were as you asked.

The push button works with the FOB, the first item to check would be that the FOB is okay and working.
Next you will need a dealer level scan tool that can talk on all of the bus systems. The push button talks to a wireless module that sends a signal down the data bus which then tells the anti-theft system to check the FOB. If it is okay it registers the push.
No real way to test if the system is seeing the button operation without a way to talk to the system.

Scanning it for operation can be done by starting it with the key, then putting the button back in and looking at the data sent on the bus to the system.

It does look like there is a software update to the PEM relating to the keyless FOB being unresponsive. It shows it can effect the "keyless go" option as well.

The first step in that is to remove and replace the battery and test it. But if it already lock/unlocks the doors and remote starts the vehicle (if equipped) then the update may not be needed.