something like that would work too. i doubt my little 10 x 22 lathe tailstock has enough gripping power to put any stress in the shaft since it is better material and decently robust in size. the large end is 2.75 inches in diameter and it steps down to 1.5 inches at the drive end. the wall thickness is pretty thick all over too.
on a bigger lathe this might be something to keep in mind but not so much on my little lathe

from what i was looking at in the PDF i posted the shaft is the most important part of the spindle. the housing has a little more give and when it comes to the drive side sizes open up a little more. basically what i got out of the PDF was that the the run out on the diameters needs to be .0002 or better and the shoulder runout needs to be with in .0001 on the run out. then the ID and OD need to be concentric with in .0002 and this is all imperial sizes for reference.
setting the second operation to do all the internal work is the hard part. you have to spend some time setting the shaft. i usually indicate the drive end first the get the far end to match then check the outer face to make sure it matches the rest. then since you need the outer end to have support you need to set a steady rest on the far end for support. i have the lathe running and an indicator on the shaft to make sure the shaft is not getting pushed around by the steady rest when setting that part up. then a final check on the 3 indicator points to make sure nothing moved.
after everything is set i put the indicator on the shaft while it's running to see if the cutting force is moving the shaft while i do the internal work. it's a process for sure
i like doing shaft work on lathes for personal use and as long as the material is not to expensive if you mess up it's not the end of the world. but i worked for an industrial pump company in the shaft cell of the plant and when the shaft is 10" x 14 ' and costs 250k that makes it stressful to say the least

but i am glad i had the time in doing that kind of work so i know what i am doing now