I took another look at the engine and found what i believe is the fuel pressure check valve. It was a stem like a bicycle tire on the fuel rail with a blue cap on it. I turned the engine to ON, then depressed the valve stem and it shot out fuel. Tried this twice and did it both times. Then, I tried to start it and after several seconds it did fire up, then immediately shut off again after running for about one second. So, I assume the system is priming, which would make me think that maybe the fuel pump isn't dead? Below is a photo of the blue capped valve I depressed.
View attachment 10432
So, is my fuel pump dead? Could it be a clog in the fuel filter? In the line? Could the water infiltration issues that killed my key fob relay be doing something? Oh, one last thing, when I began investigating the issue, I found a dead mouse under the engine cover. He had made a nest using the foam from the bottom of the engine cover. He had also slightly chewed some wires, but they are still in tact. I do need to tape them up, but I wasn't sure what effect it would have on the fuel system.
I am not sure if similar to the LR3 but I have a 2011 LR4 that had about 75000 miles at the time the Land Rover put 2 yes 2 fuel pumps on my truck and still did not help with a simile problem you described above- finally a shop (not Land Rover repair) found a sensor that was bad..and it totally fixed the problem. I currently have 205000 miles on it and still running great!