Ideally, the backplate should start by being machined to fit onto the spindle of the lathe.
If a flange fitting, first machine a register that is a tight fit on the flange. (If it can't be fitted into a chuck, mount on the faceplate, and centre as well as possible ).
Having fitted it to the spindle flange via the new register, you can then machine a register to take the collet chuck backplate.
It will be worth boring the middle of the backplate to at least the same maximum bore of the spindle, so that you can pass long work through the spindle bore..
Hopefully, when you clock the internal taper to check concentyricity, there will be almost no fluctuatioinnofnmthe needle.
Howard