Hmm ... while the Taig pie-section soft jaws will work on this, what I would think would be the best is a step collet, except that they are rare on anything other than jeweler's lathes.
Or -- some soft collets with fairly large diameters -- but those seem to be offered in 5C size mostly.
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Or -- you could *make* a form of "pot chuck". Given the size of your machine, I would suggest starting with a piece of brass or aluminum somewhat larger in diameter than the larger of your workpiece diameter and your spindle nose diameter. (They seem to be somewhat similar in this case, though I'm not sure, not having seen your lathe.)
Anyway -- in a normal 3-jaw chuck, machine it to fit on the spindle nose, and drill and tap it for the needed studs so you can mount it directly on the spindle nose.
Mount it on the spindle nose, and mark it so you can always replace it with the same stud in the same spindle flange hole.
Then, machine the ID a bit (say 1/8" to 1/4" smaller than the OD of your workpiece, and perhaps 3/4" to 1-1/4" deep. Then machine a step on the outer end just barely large enough and deep enough for a slip fit of your workpiece.
Then, turn the OD down to perhaps 1/4" diameter greater than your workpiece diameter, and down to the same depth that the inner bore reaches.
Then, remove it from the spindle nose, and use a saw (a bandsaw if you have it, otherwise a hacksaw or a slitting saw in a milling machine.
Restore it to the spindle nose (being careful to use the same stud per hole that you used when turning and boring it. Slip a worm-geared hose clamp over the OD, place the workpiece into the ID, and tighten the hose clamp to grip the workpiece.
Hmm ... you may want to drill a hole in the side of this to let you reach the back of the workpiece to nudge it out if there is not enough to grip.
Of course, you will have to make a different one for each size of workpiece, but it should give you a very repeatable workpiece holding system for this rather specialized task.
The pot chucks which I have seen in use were held in the 3-jaw chuck, but that was on a much larger machine than you have.
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I'm having a bit of trouble visualizing what you are describing here -- especially the "mushrooms". If you intend to fit it to the outside of something like a coin, without it extending beyond the thickness of the coin, what I would suggest is another pot chuck for the final OD of this, so you can bore the metal out to just too small to fit (perhaps 0.001" too small, or a bit less.) When it is made, place the coin (or coin equivalent) on a cold flat steel or cast iron piece, heat the ring to an uncomfortable temperature (which temporarily expands it), and drop it over the coin, using something like a piece of wood to hold it down around the coin until the temperatures equalize and the ring shrinks into a firm grip on the coin.
Good Luck, DoN.