Hello! Are you sure that the Immobilizer is responsible? Not only does the Immobilizer disengage the starter (keeping it from cranking), but it also disables the fuel injectors, which would prevent you from push starting it as well. Also, if the immobilizer is active, it usually shows "Incorrect key" or similar in the Saab Information Display (SID).
If its just a matter of it not cranking, but will start if you push the car, then it could be something simple like a failed starter.
If the battery has a dead cell, it can prevent the car from starting, despite attempts to charge it. I think the key fact in this situation, is that it just goes "click, click, click". The Immobilizer disables the starter (so you should have no "click") and it also disables the injectors, so that if you did push start it, it would still not start.
Because both of these are not true, I suspect the Immobilizer message is due to a low voltage condition.
What can also make this a hard to diagnose situation, is often times batteries with a dead cell will not take a "jump" from another vehicle, but I would try anyway to jump it off with another vehicle and see if you get get the "click click" into actual cranking of the engine.