Two readers, two different questions
A: Absolutely not. The shared foyer is not public in law as access is controlled and is not a place where the public is admitted either by right or upon payment. Even if it were, no offence is committed because you would have “reasonable excuse” to carry your lawfully owned gun from your home to your car and then drive to a place where you have permission to use it.
I'm a student living in the centre of Cambridge and quite regularly cycle to shoot on land on the outskirts…
Tips on how to handle a gun safely and an introduction explaining why gun safety is so important
The way to tackle this is to break your gun down into barrels and stock and put them in an ordinary holdall as you carry them out to the car. That way nobody will know what you are carrying. It may be marginally inconvenient, but if it solves the problem of someone else’s paranoid perception then it is worth doing.
Tell the FEO that is what you intend to do in future and be assertive that you have done nothing wrong.