Having been working fine on the way to the supermarket and then opening fine after being in the car park the car now won’t turn off the immobiliser. I had a car locksmith out and he confirmed key is programmed fine, transmission from key working but immobiliser just won’t turn off. Car was towed to a garage. They garage think it may be the starter motor. So my question is if the starter motor has died would that stop the immobiliser turning off? I’m not convinced!

well a starter motor fault won't exactly make the immobiliser not function but it would give similar scenario of engine not turning over much same as if immobiliser was on .
Testing starter is less than 10 minute job if your garage any good so no need guess on subject of starter functional condition .
You don't say what year you Forester is but I would guess it a SG model and has the Sigma m30 alarm system, If the alarm system is at fault simplest solution is buying a used good m30 alarm removed from a Subaru (must be supplied with a working remote or pin code so can be easily programmed to your key fobs)
Second solution is bypass/removal of alarm .
Fitting used alarm is pretty easy & we fixed a few this way and at sensible cost .
Biggest issues you face is most garages clueless and give you poor advice/work for your money .