Immobilser and locking aren't actually linked (other than they both live inside the key head).
The alarm / locking transmits when you press they fob. The immobilser has a transponder chip which only transmits when it's in the ignition lock. The ECUs that receive the signals are different too.
So sorry, the theory doesn't stack up, and as pointed out an immo lockout also prevents cranking.
So cam sensor, fuel pumps, or glow plugs. If it's OK when cold, but always fails with a hot engine, I'd plump for cam sensor too.
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