The ultimate test will be live fire. All the work needs to be done to the front half of the gun. If any work is done to the back half the other barrel assembly will no longer fit properly. I think under firing the action slide will override the action slide lock and lock up the works. In any event it is not an ideal situation. 1 mm is 39 thousands just enough play to foul things up in gunsmithing. I think working up to 39 thousand by removing material from the face of the barrel chamber area would work. But as stated this is machine work not dremel or hand file. With the sleeve removed and the barrel trued and leveled you may be able to remove that much with careful sanding on a disc sander. A little backwoods but it may work. Sand and fit sand and fit sand and fit goes the plan.
Darryl
Yeah for sure.
I would take the approach of the trying to minimize the insert myself. I'm a backwoods type of guy though!

Would be slow methodical and painstaking but it could theoretically be done at home with enough care and the right tools.
Milling the face of the barrel/mage tube guide down would be a straight up gunsmith job. I would not even consider it with my capabilities.
OP; I would confidently fire a live shell with that little play if it was my gun but don't take that as a "go do it!" statement for yourself.
Would take it to a range area, try cycling some live shells in a safe direction and see how the chambering, extraction and ejection works using the unlock lever behind the trigger.
If that all went smooth I would fire one live shell. Again carefully pay attention to the cycling of the gun. Once shell is extracted, inspect the base of the of the shell for bulges or obvious signs of head space problems. In a shotgun you will be primarily limited to case head separation/cracks or ruptures of the copper/brass base
Any hard jams in the extraction or headspace pressure signs and you will have to address the issue.
With that minimal gap and it being a straight wall shotgun shell chamber I would not anticipate any issues.