I take it that the FPR used to hold pressure? Did you check the FPR after you installed the fuel pump? Some fuel pumps will allow the fuel to spill back into the tank allowing the pressure to drop.
If you try to start the engine immediately after you turn the key (before the fuel pump stops) does it start any better? Does it start at all? I probably should have asked that first, but I get the impression that it does since you are concerned about burning out the fuel pump.
Do you have a hand vacuum pump that you can use to apply vacuum to the FPR? If you do, hook it to the FPR and apply vacuum while the fuel pump is running to see if the fuel pressure drops as you add vacuum. Fuel pressure should hold steady as you hold the vacuum pressure steady. E.g. If you apply 5 psi of vacuum, the fuel pressure should drop 5 psi and hold constant as long as 5 psi is being applied. If the pressure doesn't drop, replace the FPR. That means you probably have a bad diaphragm.
If you don't have a hand vacuum pump, what happens to the pressure as you crank it over? Does it drop or stay constant?
Just some ideas.
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351W 89 Mustang GT Convertible