A machine lathe might be the tool of choice here, but here's a notion of how I might do this job without it. First, two things: You need to hold the piece solid as a rock, and use a suitable guide bushing for your drill bit.
The bushing must be referenced absolutely on the work-holding fixture and be immovable during the drilling operation.
For the work-holding fixture, I envision turning a tapered recess in a very stable block, maybe laminated plywood or something that isn't going to move on you. Without knowing if your pieces are uniform, but assuming they are all circular in section, a tapered hole would have to work, but it might take some trial and error.
Then I'd rig up a housing for your drill bushing (e.g., https://www.mscdirect.com/product/details/07124050) that would be registered absolutely concentrically with your holding fixture , probably with two or three tapered pins, so the alignment is repeatable. Use a 3/8" x 6" twist drill in a drill press, with your work-holding fixture securely clamped directly in line with the bit. Test it first, making sure that that the bit is perfectly centered with the bottom of the tapered hole in your fixture.
If all this fails, try Keith's notion. It's in keeping with the latest and most forgiving hardware theory, where you drill oversize holes and let epoxy fill the voids.