I would suggest using the lathe is the eaiser option because to use the milling-nachine as you suggest, may mean having to set tthe shaft etc in the "corner" off the table.
I have done that for one or two awkward tasks, and it is not easy!
You mark the cra nk end as having weard bands but are there bands between them on which you can place the DTI? I am thinking of being better to set the steady at the chuck end then slide along to the support point.
That of course assumes a constant diameter but I see the outer end both steps down and appears to have a groove extending from the key-way, and which might create problems for the steady unless it is much narrower than the steady's contact areas. I think if you can mount the steady right by the step you might be OK with very careful machining, but as John suggests, putting a bush on the shaft would be better.
Obviously you can't face that end back more than enough to just clean it up, but I would try using a small knife tool to ease the damaged area of the centre-drilling down enough to give re-drilling it a better chance of concentricity. It doesn't look as if the shaft is especially hard there so HSS tools ought cut it.
'
Incidentally you learn something new every day here! I have never previously heard of a "spring dog" yet lo and bohld, there is one, on the drawing of the lathe set-up. As the "words and music" bear out, it holds the work back onto the live centre, since the outer end is not constrained axially.