Sentry Key Remote Entry Module. One of its functions is to immobilize the vehicle if the ignition key fob transponder chip isn't recognized. So if the module itself became faulty it would no longer recognize the valid key fob. But before you even start investigating the SKREEM module, let me ask you again.....

Have you verified that the relay control module (RCM) is not faulty? The reason I suggest this is that you mention in your original post that the engine started and then died quickly twice and then would not crank at all. I had a similar experience and it looks like it was a failing RCM relay (and not the more common solder problem). Before eventually becoming a never crank situation, I experienced several starts followed by quick stalls or just cranking with no start. The RCM is plug-and-play so if is the problem it can typically be repaired or replaced rather inexpensively since you don't need to program anything (unlike when replacing the PCM or SKREEM). I'd recommend removing the RCM and inspecting it if you haven't already.