I searched and found this thread as I think I'm having a similar problem. I recently purchased a 1996 xjr. I had it shipped directly to my local shop while I was in the hospital and spent 7.5K. Tires, shocks, hoses, belts, fluids, etc. No engine work as I saved that for me. I compiled a list of things to order for the engine based on my reading here. In the mean time I drove the car. Started with a slight miss. On the third 2 hour round trip I barely made it home. Barely running and used a full tank of gas in 150 miles.

Based on everything I read here I suspected oil in the spark plug wells or bad coils. The wells were dry and as part of preventative maintenance referenced above I replaced all the coils one at a time to see if that made a difference. (NGK coils.) No difference. Checked crankshaft sensor. Looks new and clean.

So then I was thinking fuel. Pulled the vacuum line from the FPR, dry. Then while running (barely) I pinched the fuel return line and voila! the engine smoothed out. As testing the fuel pressure appears to be difficult I went with this method and ordered a new FPR. I just now installed it and no dice. Same problem. Pinch the return line and it smooths out. Note: On start up it runs perfect for about 3 seconds. Then all hell breaks loose.

At this point I think I'll jumper fuel pump 1 and see what happens. Then maybe pump 2 if that doesn't work. Depending on the results perhaps some of the other tests referenced above. I'm really hoping I don't have to replace fuel pumps.

I live at a marina and the car is in the underground garage. (Cold!) At this point I'm hoping to get it running well enough to drive it to my shop 25 min away where I can do all the other work. I have to cross the Chesapeake Bay bridge so I need to be reasonably sure it will make it. I called and a tow truck is $250. Just enough for me to try and get it running well enough to drive. I have trucks and trailers etc but it's so tight where I live I'd have to do it at midnight when no cars were parked on the streets. $250 doesn't sound so bad then!

 

there was something about if the engine does not start after a certain point the ECU will command the # 2 fuel pump in the starting sequence

once this happens there will be no second fuel pump above 4000 RPM

A quick try would be to remove the # 1 fuel pump relay

As the fuel pumps fail they will pit the relay power contacts inside and burn the sockets on the pump motor connector

In my area the local small equipment rental vendor has a tilting flatbed car carrier by the mile that a driver is supplied at a much cheaper rate then a wrecker operation

Its a place where you would rent a portable cement mixer and like items for personal usage and not necessarily commercial

 

The T fitting for piping into fuel pressure comes from Australia.

What I did for mine was drill a hole in the upstream side ( fwd ) of the fuel pressure regulator on the hex flats , not on the fuel rail

J - B welded in a thin tube from a hobby shop hooked to a tire pressure gauge lasted long enough to give me the 43 PSI

Eventually replaced the FPR.

the # 1 fuel pump on yours follows the failing pressure / volume relationship as it must trade one for the other as it can't satisfy both

in the starting sequence the # 1 runs for 3 seconds only ( ignition switch ) at the moment the injectors fist draw little on the trapped fuel pressure

as the pressure is drawn down the pump will come back on when the CKPS sees certain engine rotation

This second " on " will either be full on or full off and no in between pressure.

Probably a good idea to open a thread on good fuel pump brands as there are some bad examples.from reading.

there is a check valve on the new pump that can be bad out of the box , this traps the fuel pressure in the start sequence.

in your case with 2 fuel pumps either pump having a bad check valve effects the whole fuel line system