Great ideas! I had considered using my faceplate, which is plenty large enough to accommodate the raw pieces, and I have a tool I made for another project that's basically the geometry you describe, except made from a larger blank. Hmmm...looks like I'm going to spend more time trying out new ideas than actually making the parts! But that's a very good thing, if you ask me.Without knowing the rest of the part geometry, it is hard to say what would be the best approach. I would consider using a faceplate with hold down clamps and a sacrificial backing plate. If the hole is predrilled with a pilot hole. the hole could be located using a center in the tailstock and clamped down. If a faceplate isn't available, a plywood plate could be mounted in a four jaw chuck and the work fastened to it.
For machining the hole, I have done trepanning using a 1/4" tool bit ground to fit the purpose. There needs to be sufficient relief to clear the diameter of the hole but 3/16" is a piece of cake. Far easier the machining out the entire material in the hole. (Bonus, you end up with a 3/16" thick washer blank which can come in handy on other projects.)
View attachment 310019
Thank you, sir!