2003 360, immobilized light not working

So your fob is operating the doorlocks and your car runs?

Assuming the light is plugged in (check the back of the panel)...

The light is supplied directly with power by the immobiliser ECU, so either the wiring is faulty or there is some part of the Immo ECU not working. LEDs usually don't fail. Maybe look for voltage going to the lamp with a dc voltmeter. I can't tell you if it's 5 or 12 volts (pulsing), but if you get either, then I'd say the circuit was ok. Wiring continuity checks from the ECU to the lamp might be a little more difficult as the lamp wiring goes to the bottom plug of the ECU and the ECU is buried behind other components.

 

Maybe the immobiliser battery is dying ? My recollection may be faulty but doesn't the system still work even then

 

Are you talking about the fob light or the dash light? When you say you pulled the fuse cover off, I assumed you were talking about the car as there are no fuses in the fob.

 

Second vote for failed siren.