Join Date: Dec 2014
Location: Rocky Mountains, USA
I would think you could part down to a lot less than 1", maybe more like 1/4" or even 1/8" if you have a light touch, before the turning becomes unstable between the centers.
I'd do the tailstock end first, then the headstock end, and keep the tailstock pressure as light as possible consistent with stability.
Even if you do overcook things a bit and the workpiece comes unmoored from between centers, in a spindle turning orientation like that the outcome is more irritating than dangerous. The work might zip around on the floor for a bit and worry the shop cat a little, but that's about it.
A 1/4" nub is going to sand off a whole lot quicker than 1" one, especially if you quickly pare most of it away with a razor sharp chisel first.
There are other ways to hold work like that in a production situation, but I don't know what equipment you have on hand (like chucks for example), so I hesitate to get too deep in the weeds with other suggestions at this point.