You are correct in suspecting the immobilizer (with the flashing green key), but before tinkering with with the immobilizer itself, check 2 things:
- is your key the issue? Chip keys and transponder keys get worn out. While the key may be mechanically fine, sometimes the chip gets partially disconnected. If you have a spare chip key for this vehicle, try it. Also, some transponder keys have batteries inside them, so replace yours if you have one. You may also want to remove the key from your keyring, most people have way too many things on there and it's pulling the keys and the ignition down, and possibly interfering with the immobilizer seeing the key.
- tried the remote start? Immobilizers prevent you from driving the car without a legit key, but you should still be able to start the car with the remote. You won't be able to drive it without the immobilizer seeing your key, but this will rule out a few things.