What is the cost to fix a weak fuel pump on a lincoln LS 2002 v8

do a fuel pressure test. Sounds like a weak pump or faulty fuel pressure regulator.

located inside the fuel tank with the fuel pump.

It's possible your crank sensor is going bad. Sometimes they won't light up the engine light. You can drive the car with the diagnostic tool sill hooked up while driving the car and when it dies the computer will show the code. You have to do a drive test.Check the EGR valve also. Hope this helps.

The inertia switch should be in the trunk behind the trim panels.
You may need a pump relay.

If the alarm light is blinking, then the issue could be with the vehicle immobilizer or key. The key may need rematching to the vehicle's ECU, which will need reprograming by the manufacturer service centre.

Someone would need to check the fuel pressure with a gauge before and after you start it.
Could be the pump relay or the check valve inside the fuel pump in the tank.
If the check engine light is on you would need to get the computer scanned for trouble codes.

car is equiped with two fuel pumps, a transfer pump and a high pressure pump, book time is 2.5 hours to replace both pumps or 2.0 hours to replace one pump, the pump cost is $ 333.04 for the transfer pump and $ 475.75 for the high pressure pump these prices are list price at NAPA.

$ 475.75 high pressur pump
+ $ 333.04 transfer pump
+ $ 125.00 Labor ( $50.00 per hour X 2.5 hours )
------------
$ 933.79 + Tax

If the fuel pump is new then it could just be your fuel filter which is a cheap fix so id try that first. Or it could be your fuel injectors.

How did you test the fuel pump?
Which one Distribution Pump or Transfer Pump?
Do you have fuel presuure at the rail?
Do you have a crank signal or spark?

V8--V6!

does it have a crank sensor? could be on its way out. even an ignition module could cause this to.