Since I have a lathe and tig welder in my shop I'd personally build it up then turn it down. Would want to take some inspection
time at the end to ensure I didn't pull the shaft with the heat from the weld.
If the shaft was clean and ready to weld on I can't see the whole operation taking more than an hour. I'd first set it up between centers or a collet if I had the right size on the lathe. See what kind of runout you have. Then undercut the worn area to get to clean metal. Weld it, turn it down within 0.001" of finish size then polish it to final size. Check with indicator to see if runout is in spec. If not use some spot heat/cooling to pull it back straight.
It's very common to do this for shafts outside the marine
world, large electric
motors, pumps routinely have this done. I'm not a pro so I can't tell you the answer to this but there is a limit to how small the shaft can be or the heat from welding on it will warp the shaft where it isn't feasible to do this procedure.
So I'd say if you brought to a machine shop, maybe a few hundred dollars max as a guess. I don't know where you live or what the going rate is. Or what a new shaft costs.
There's a guy on YouTube, Keith Fenner. His company is called turn wright machine works. His specialty is marine
shafting but he does stuff like this all the time. He's a true old school
machinist. I knew him before but not well. I was on a J40 the night before the start to Figawi and the boom snapped in half. Like completely and utterly snapped in half. I called him up Friday Memorial Day weekend at 6pm sailing into Hyannis. By 10 pm we were driving back to the boat
with the boom completely and expertly repaired. Skipper
is still running it that way 2 years later. Might want to give him a call see if he'd take it on for you.