Didn't you say you replaced the PCM and BCM? I had issues with and AUDI that it would do the same thing. A bad ground was the cause. Starting would work, would fire, then run a few seconds and die. The ground would lose contact and cause it to lose signal and stop running. May also be related to that battery issue you had. May have shorted the PCM internals.
Sorry no update yet, got sick yesterday and left work at 11am, went home and slept for 16 1/2 hours. So tonight I will probe the pcm with a multimeter, check volts and ground, and replace ignition switch