@XWrench3,
I have done this, but to much larger and more expensive axle housings, but it's been some thirty years ago. That $500.00 just might turn out to be a bargain, but anyway, stainless is not required, nor is heat treatment. I do think you're going to have to turn it down more than .060, as this will leave only .030 on a side, which is quite thin.
Now, I'm speaking of the axle housing, not the axle itself. If this isn't what you are contemplating, you can pretty much ignore the whole post. Anyhow, if you are setting the housing (including differential housing) up in a lathe, it's important that you don't center the off end in the chuck. You'll need to offset it so you don't lose the camber in the damaged end. Just a tip in installing the sleeve, if you go ahead with the project, we used to wrap a piece of steel banding strap around the sleeve, and hold the ends with a pair of vise grips so we could heat it for a good shrink fit.
You can forget a whole lot in three decades, so I hope I haven't forgotten anything vital. By the way, most of the axle housings I reworked came in with the end completely broken off. If you've just got a groove, that should take much less time. Again, $500.oo doesn't sound all that bad.