First of all, I would use a smaller spur drive, such as a half-inch drive or a ring center (see link below). If you use the Steb-type drive shown in the photo for safety's sake, the ring center provides the same level of safety and drive capabilities without leaving big marks or needing to leave nubs at either end. I suspect, also, using this set-up would save sanding time - the parting tool leaves a very rough, torn-out surface. If the stock was originally cut clean at the ends (miter saw, table saw, good bandsaw blade), the ring marks and teeny-tiny hole from the ring set from Axminster (or others?) would be easy to sand out.
If you prefer the drive and live centers you have, try leaving longer nubs, bevel the nubs down ward toward the project until there's a very small amount of wood left, face off the ends with a sharp skew rather than a parting tool and proceed to 4emove the nubs