Most press setups Ive seen to print on round things put the round object horizontal on rollers. The screen is set to a slightly lower tension. It lowers down onto the object (along with squeegee) and slides across the object. This motion effectively turns the object on the rollers and prints (in the round). I'd also suggest sticking with one color. Keeping register is going to be a royal pain in the ass, esp with a stencil setup. I can also see what coy suggested working. That is providing the circumfrence of what you are printing is not too great. Otherwise you just won't have contact with the screen.
I kind of like the idea of a curved frame, but that poses all kinds of callenges in itself. I'm not sure how you would get an even coating of emulsion on either, unless you used capillary sheets, which are pretty expensive.
Oh yea, and my pet peeve - its NOT silk!!!!! Most screens are made of some sort of polyester, or blend of other materials. Most cheap screens you find at craft stores can even be cotton. I guess some people doing fine art reproduction might be using silk, but its pretty rare [/rant]