.... I'm afraid this is going into the wrong direction.
First of all, the Injector is supplied with 12V permanently when ignition is switched on. The ECU triggers the Injector to ground, when signals from crankshaft sensor are received by the ECU.
If you don't get signals from crankshaft sensor, the fuel pump does not run when starting the engine and there are no sparks! Easy to check.
The immobilizer has no connection or influence to this! So please check why you can't measure 12V at the Injector when ignition is on.
Other things:
The Fuel Pump gets a signal for priming the system for about 3 seconds.... not 10! If the Fuel pump is running for 10 seconds, there's something wrong.
If you can engage the starter motor, there's nothing wrong with your immobilizer. No need to change any wiring.
Injector is No. 87 in the upper right corner.
