You are looking at buying new, in the $80-125K range for a suitable, capable lathe. Options I would not consider optional, would be a parts catcher, and a bar feeder or puller with a pneumatic collet closer. Another $10-50K to cover tooling, software, shop supplies. You are going to need some accurate measuring equipment and a temperature controlled room, to keep it in, if you expect to be able to reliably say what the actual size of the bullet is. At the least, an decent bench micrometer, and an assortment of gage blocks to check it against, as a start for the measuring equipment. An Optical Comparator is going to make seeing what is wrong or right with the finished product, that much easier.
Or somewhere in the $5k range for a used Hardinge, accepting that it is a crapshoot that it is still in good enough condition, close to that again for tooling and supplies to get started (everything from cutting oil for the lathe, as they are not set up to deal with soluble coolant, to some bar stock and spare carbide inserts).
Trying to do this stuff on a manual machine is not impossible, but it will get to be not fun in a hurry, followed by feeling like torture, then hatred develops, and you don't want to be near the stuff any more.
Trying to pull it off with too light a machine, will make you wish you stabbed your eyes out with a fork, as you will be chasing tolerances back and forth all day.
Have you considered just farming the work out to someone that is suitably equipped? Not me, by the way. I'd be looking for someone with a CNC Swiss lathe, that was looking to keep it running, if I were going down this road. At least for the beginnings of this adventure.
The price quotes that you get back, at least, will give you a really good idea of what the cos of these should be, in real world terms.
Standard fact of life. Decimal places in the tolerances add decimal places in the bill. If you really want to spec them to .00000x, be sitting down when you see the quotes, eh.
Cheers
Trev