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Polishing inside a groove in a carbide ring

Hello,
Being a mechanical engineer and a tinkerer I am making rather than buying an engagement ring. After much thought I settled on tungsten carbide due to its scratch resistance, and because I was unable to find any WC rings commercially available in the design I planned. The rough shape of the ring as seen from the side is a V (i.e., two rings at the top, becoming one at the bottom) with three stones emerging from the gap at the top of the V. So far I have obtained the blank, rotary ground it to the correct OD (ID was already correct), cut the V shape, as well as an internal groove (to help secure the metal which will hold the stones) on a wire EDM, and rounded all of the edges with a green SiC wheel on a rotary tool.
I have several grades of diamond pastes (5 g each) in the mail.
My problem is that I do not know how to polish inside the center of the V. The OD/ID I can get to, but the V in the center is less than 1 mm at the top, and goes to a 0.006" radius at the bottom.
I considered buying or building a tumbler, but am unsure 1) if it will work and 2) what medium to use. I am aiming for a mirror finish. Should it be required, a very long run cycle will not be a problem.
Any and all advice is appreciated.

Daniel Sing
Hobbyist - Indianapolis, Indiana, USA
December 28, 2009