This sounds very interesting, but how do I get repeatability on the right side of the flange?
Suppose your mandrel is RH-threaded from the middle toward the tailstock, and LH-threaded from the middle toward the headstock, with a raised portion between the two threaded sections. The raised portion is larger in diameter than the RH or LH thread major diameters, but smaller than the part OD. The female-LH-threaded flange is faced off flat, with a recess that accepts the raised portion of the mandel. When the flange is advanced toward the tailstock, the raised portion of the mandrel is a positive stop for the flange. When the part is threaded onto the mandrel, the flange is the positive stop for the part.
Under cutting pressure the part and the flange will both be screwed more tightly to each other because the threads have opposite hands, but neither can move because the flange is stopped against the raised portion of the mandrel. Because the flange can be made with arbitrarily large diameter and convenient places to grab, it's easy to back off the flange after making the part. That releases all the tightness on the part, so it's trivial to unscrew.
Basically, the flange is an easily moved shoulder to locate the part, and the raised portion of the mandrel gives repeatability to the flange when it's replaced.
This might be a bit hard to bring off at 5-40, but it was a pretty common approach in the era of manual turret lathes.
There's a variation which may be more practical, where the LH-threaded end is much larger in diameter. In this case, there's no way to have a raised part of the mandrel
between the two threaded sections of the right diameter. So you would use a flange stop mechanism placed toward the headstock end, instead of in the middle. For example, cut a broad groove in the circumference of the flange. Fasten an arm to the mandrel shank with a pin that sticks into the groove. The width of the groove limits the travel (in either direction) of the flange. You don't need more than a turn, maybe 1/2 turn. This lets the RH-threaded portion of the mandrel be small, weak 5-40, while the LH-threaded side can be much more substantial to grab in a chuck, etc. Make a disk roughly the same diameter as the flange with a bore just large enough to pass 5-40 threads, and fasten it to the tailstock side of the flange. That provides the surface your parts will locate against. The arm/pin/groove will locate the flange at a repeatable work position.
If you imagine a flange with a T-shaped cross section (rather than rectangular), the groove can be cut in the reduced diameter portion so the arm isn't flying way out there. And you can make the arm balanced, as a tube or 2 or 3 arms equally spaced around the mandrel circumference, so you don't have dynamic balance issues.
... This is all easier to describe with a few drawings ...