Buck Algonquin may have been the original supplier of the shaft log, but in any case they have new replacements available. Not that big a deal to replace one, but it will take a haulout, removing the prop and pulling the prop shaft back. Buck Algonquin also sells the heavy hose that attaches to the shaft log which after 30 years needs replacing too. Use double T-Bolt clamps on the hose.
I am surprised your mechanic doesn't know this. And FWIW, welding even if it might be successful which I doubt, requires doing all of the above work to get it out. The cost of the new part will be a small fraction of the total bill.
David