I was surprised when you reported the fuel delivery problem was fixed by the installation of a new battery. I was not surprised when you reported the problem returned - if it's indeed the same problem.
1. Are you using a Battery Tender to keep the battery charged when the car sits idle for more than two weeks? If not, it's worth considering.
2. Has anyone scanned the car with a Tech 2 to observe what's going on with it? I ask because you've thrown an awful lot of money at the problem that could have been more enjoyably spent on tier-one beer, and the problem remains, waiting to frustrate you further. Playing troubleshooting "Wheel of Fortune" with this car is a losing proposition.
3. The recently-replaced module might have failed, but without a full parameter scan of the fuel system, it's difficult to troubleshoot remotely. The Tech 2 takes a lot of guesswork out of the troubleshooting equation.
CCC