The difference, as I see it, is you're comparing a low pressure round (shotgun) to a high pressure round (357 Sig), but using the same basic concept for both.

The fit of the barrel liner would have to be so tight as to make it hard to install and remove. The difference between a .400" diameter bore and a .356" diameter bore is only .044", which means the maximum the barrel liner walls could be is .022", which doesn't leave any room for clearance between the inside diameter of the 10mm barrel and the outside diameter of the 357 Sig barrel.

The other problem is the shoulder of the case. Despite what you read about the 357 Sig round headspacing on the case mouth, there is the second datum point on the shoulder to consider. If the shoulder of the case contacts the shoulder of the chamber insert too soon, the slide will be held out of battery. At the opposite end, if the shoulder of the chamber is too far forward, the case shoulder will blow out to match the chamber. This probably wouldn't result in a case rupture, but brass fired in that chamber would be excessively worked and would fail prematurely, in my opinion.

The idea is novel, but I just think it's not applicable in this case.

Hope this helps.

Fred