That test while driving points to a problem. When you hit the gas, intake vacuum drops, and that feeds the FPR, which tries to raise the pressure. You said it goes up while in park and the vacuum line is removed from the FPR. Those two tests tell me the pump isn't pushing enough volume to keep up while the engine is running faster and under heavy load.
Don't measure vacuum at the FPR, that should be a pass fail device. Now if there were a tiny leak internally that bleeds off some vacuum, that could be an issue. But in general that would be extremely unlikely, super rare, and isn't creating the fuel pressure drop. It's likely the FPR is perfectly fine, they very rarely ever fail.
But the pumps often do, as well as the at pump rubber hoses, sometimes develop a leak or split that lets fuel out. If you aren't sure of the tests done, double check the in park test again. Pull the vacuum line off of the FPR, and if it jumps to near 40psi quickly, it's the pump.
I'd say you need to work on R&Ring the pump, and carefully check the clamps and lines in there too.