I'm with Pete on all counts. While peening is looked down upon, it is somewhat effective. All traces of peening can be removed later during a weld and refit, if desire. Meanwhile, I'd use it with pleasure until it loosened up, if ever.
If it then needs work, I'd opt for the weld and refit over the shim method. But shimming has advantages for those that want to do their own fix and don't have access or experience to do the weld refit or don't want the expense.
You may want to determine if the barrels are damascus or homogenous steel. Also, the proofmarks may give some valuable info. Some pics of them could wrangle some good feedback from those knowledgeable in proofmarkings.
