I knew one of the best welders ever, gone now, but has stuff sitting on the Moon that he welded. For a job like this, he would use a coathanger for a TIG filler rod, tack it enough to hold, but not too much not to bend easily. Then he would set up a dial indicator, and make sure the gear was true all around, tack it a couple of more places, test it, then weld it with a regular filler rod once it was okay, and secure in place.
https://www.youtube.com/watch?v=SZjZJSig4bo
Since that shaft would need to be replaced otherwise, there really is little to lose, if you choose a good welder.