While the ecm powers the lift pump it should not care where the power is coming from. I seem to recall that at "key on" the ecm runs the pump for 5 seconds, if you bump the starter it runs for 30 seconds. During the normal start up sequence it runs for 5 seconds then when you move the key to start it has full power. The lift pump does not care where it gets power and as long as it is wired correctly should not pose an issue to the pump.
Don't know where your pressure gauge is plumbed but if it is post filter could be a dirty fuel filter, could be a faulty gauge, the fass is getting weak for whatever reason, could be the pump return pressure mechanism (spring I think) is going south as well or junk in the pump not letting the bypass operate correctly. Could be air getting into the system. Could be a bad overflow valve on the VP though that is fairly rare that they really go bad (iirc a new pump does not include a new overflow valve).
Not in any specific order I would break out the multimeter and verify that the pump is getting 12 volts, check the fuel filter, verify the fuel pressure with another gauge, look for fuel line/connection air leaks. A variety of other things could be an issue such as delaminated fuel line pretty unlikely but has happened, junk in the tank clogging the pickup screen.