One could certainly simply leave the shim loose on the action, just being aware of it being there while cleaning the barrels.

The notion of soft brass or aluminum seems good.

An article in DGJ, describing an identicle brass shim repair, free of any solder or adhesive, by a gent who did just that with a Nitro Special, and used the well worn piece as his waterfowl gun, with a steady diet of big loads for many years leaves little doubt the repair is good enough for a working gun.

Best,
Ted