No luck with the picture.
Most bike shops would have have repaired punctures and changed tires, particularly with Brompton since taking the rear wheel off is a pain, and many owners would rather bike shops did it. If take your bike to a Brompton dealer, you would expect them to know about adjustment, which isn't rocket science, after all. After all, you expect a Renault garage to know how to service a Renault.
The good news is that, assuming you use Schwalbe Marathons, you're not likely to have a heap of punctures in any given year, and Sturmey Archer seem to be pretty resilient with an albeit small range of adjustment. Mine protudes out closer to 2 mm, probably because the adjustment chain isn't screwed in particularly tightly.
In any event, that 1mm is a rough guide. My clone doesn't have the inspection hole, so the only way to adjust is through trial and error.
I take your point about bike shops. They tend to be small businesses here, and even sports chains like Decathlon tend to have one specialised bicycle repair person per store. There are only a handful of Brompton dealers - my nearest is close to 200kms away.