Turns out, chamber sleeving is something that is done for time to time so my thought is not too far fetched.
Ltk, chamber sleeving is IMHO, definitely an option.
We aren't talking about super high pressures here.
If a C96 Mauser Broomhandle pistol upper can be resleeved to accept factory pressure cartridges, there isn't any reason it wouldn't be safe to make up similar sleeves for a SxS shotgun's chambers.
I'm not going to offer to do the job, but it's interesting. I have sleeved several 22lrs, 22 WMRs, and 30 M1 Carbine stubs to revive shot out bores, clean up chambers, or just to fit a tighter chamber to an otherwise good bore
Sweating in chambers is an easy option for a single barrel, but in the case of your SxS, there are many other issues to be dealt with.
I wouldn't try to separate the barrels from their block. I would look for a way to ream or drill out the chambers, make up a sleeve and use a high tensile epoxy, such as Titanium Putty, to take up the few thousandths clearance, to hold them in place.
I would ream the inserts before installing them. This would mean knowing the exact dimensions of the opening, and machining the insert to fit.
The next thing to figure out, is ream before machining the OD of the insert or doing it afterwards. Likely, I would ream a chamber in a steel cylinder, then machine the cylinder to fit the hole.
One thing to think about is the present chamber has cut outs for the cartridge rims. I would use the outside diameter of that to determine sleeve wall thickness. This will make a lot of things much easier later on, such as extractor slots, and would make the repair almost invisible.
Then, you would also have to make up a mandrel to insert into the sleeve, to keep it aligned properly with the axis of the bore of your barrels.
Much more to it, but IMHO, your biggest problem would be getting a drill bit or reamer matching the diameter you need.
I haven't looked online, but someone may even be making up the inserts you need already, and all you need are the right reamers/drills.
You may even be able to adapt choke tubes, not likely, but something along those lines, without threads.