The baritone mouthpiece I really like has a small drawback - the shank bore is huge! Its not a case of someone boring it out - its a 'hybrid' with resin body and brass (silver-plated) shank tube, like a Drake. My bari has the Yanagisawa-style smaller-dia. neck. I played a couple of Runyon Quantums on it at first but eventually tried this Barkley 'Hybrid Jazz 8' that does the trick on this bari. I just added a layer of cork to the original cork to try the mouthpiece and I left it like that. I tried a different mouthpiece the other day with a small shank bore so I had to remove the extra cork to play it. That went okay but not as great as I had hoped, so the mouthpiece went back. That left me in an unplayable condition, so I got my stuff out and proceeded to put a new cork on it with the thickest cork I had, probably 2mm. Turned out nice but still a little too thin, but I can use it with some Teflon tape.
So, I ordered a selection of cork up to 1/8". I think the 3/32" will do it. Anyway, the shank I.D. is .745" or 18.92mm. That is essentially the bore of a 12 gauge shotgun!