Good luck in finding the solution to your problem.
I was doing a bit of an investigation today on the operation of the fuel pump, it may not be relevant to your problem only tangentially.
The fuel pump is activated by the key being turned to the ON position and is timed to run for some seconds by the TCM at which point it is turned off. To start running again the pump needs good signals from the CKSP and the CMSP sensors. The pump then runs continually while the engine runs, excess fuel being returned to the tank.
Like you I would change the battery as well, I figure four to five years for a battery. Mind you my last one lasted less than two years and died in the winter hibernation even though hooked up a couple of times to a tender. The OEM Valeo still holds a charge and it too is kept charged, it was made in 2004, amazing.
Personally I would look at the connector to the CKPS for corrosion, no moving parts in it and the only thing that can happen I guess is the coil insulation fails.
Remember the CKPS is removed easiest by removing the screw before undoing the connector. Putting tape into the socket so it is a tight fit I hear is a good idea

Again good luck.