.002" tolerance, which is .05mm where i am from should be a piece of cake. A .1mm (4 thou) taper, is not too bad. Its a parrallel pin so that makes life easy. if it had a few changes in diameter it would suck. What I do to remove taper, which is spot on within .01mm usually, is measure from the tailstock end and mark along the shaft every time the diameter changes (hopefully up) towards the chuck .01 of a mm. If its going smaller at the chuck its a bit more tricky. Anyway, then with the machine off, move the saddle until the tool is over the first mark, mark somewhere on the bed where you can see, i use the leadscrew or the vertical flat of the bed underneath the vee way, so you know at what point the diameter changes respective of the saddle position, not the tool. Obviousley you cant move the top slide otherwise your markings will be out. Repeat for all marks. When turning the tailstock end diameter is your proper diameter you use for tool adjustments. Everything else is bigger or smaller than that and is affected by your taper correction only. So when your turning along every time the saddle approaches the mark you tap the tool in .005mm. This works pretty well, will suffice many times over for you. Then when your done run a bit of emery tape over the lot just to smooth it up. If its needed. A .005mm step isnt easy to feel but if your finish is fine, it will be more noticeable. You need to mark the bed in this case so you know how the lathe cuts. If you mark the shaft only you will cut your markings off and wont be able to repeat the cut again. This way you can. Over and over.
If the diameter gets smaller at the chuck, ideally you want to reverse it and cut from the chuck out. Otherwise, cut from the tailstock like normal, and pull the tool out. Allways use a dial indicator for measuring your adjustments as the dials even with the backlash taken out are rubbish. Pulling the tool out has its issues. You will need a fairly stiff cross slide to prevent the tool pushing the slide out as the screw is not pushing against it. Being 4340, might be a bit tricky. Can be done but. Might have to tighten something there.
Other factors that will be affecting your taper:
If the taper is constant from one end to the other adjust the tailstock either in or out. If its bigger at the chuck end, tailstock adjusts away from the operator. If the taper is constant, but its the bed wear only, adjust the tailstock anyway. Will get the job done
. Not good, but it works.
Live center: make sure the live center is at operating temperture. A good lathe can cut spot on first thing in the morning. Horrid 10 minutes later, and spot on 30 minutes later when the center warms up and hasnt bent off center due to expansion etc. Does happen.
Tool pressure. Thats a decent size shaft id be putting a 1/2" center in it if its possible. The tool may be pushing the shaft away at the tailstock end. Try using a smaller nose radius with the bigger center. Shorten your tailstock quill right up as much as possible. Rigid is the word here.
Anyway taper corrections work well if you work out where its coming from in the first place, im sure with a bit of work it will come easy for you. I hope the diameter doesnt get bigger than smaller as that gets really tricky to adjust
. Backlash - do i need to explain lol.
Anway im not keen on using imperial for this. IMO get a metric dial, they are way more sensitive, or a finer imperial one. You can easily split a .01 division on a metric dial.