I don't know how the immobiliser circuit is integrated to the injection system, I thought it was separate. Anyway, injectors on 3.9 Hotwire EFI are as far as I know earth switched. There was similar thread not that long ago with the same problem, turned out to be the EFI loom. If you do a search I'm sure you'll find it. Carefully check the loom for continuity and earth. It might also be worth just making sure it isn't an ECU issue by testing by substitution.
Check and double check all the earth points, make sure they are all clean and tight.
Also, as far as I know, the 14 CUX ECU that was used for hot wire injection systems did not change from the first hot wire Range Rovers so you should be able to use any 14 CUX ECU but be aware there are at least 2 or 3 part numbers for the same box of tricks. Let us know how you get on.