I think you may have mixed up the scullery with the laundry Kaldin.
The website clearly states scullery. You will see from Jeremy's sketch that the scullery does not have an external door.
Yes, it does say scullery, but people seem to call different rooms different things. I'm going by the settle as well. If Jeremy had been out with the gun it would be more logical for him to leave it in the room he called the "laundry" as that is near the back door. As you say, the scullery has no external door.
This is a picture of the scullery.

Maybe Mike knows exactly where Jeremy said he left the gun.