I agree with the idea of using a 3D printer to make adapter fixtures. This helps your ONE tool accommodate more "round" stuff". I also use common Delrin on my lathe to make simple fixtures.
Some ideas for how to keep the glass from wobbling:
1) mix some sand and play-dough. Put it into a small bag. Press this into the bottom of the glass. The play-dough should form to the glass and add the desired weight. The plastic bag is intended to make it easier to remove.
2) 3D print a lever-spring system. I designed a contraption with a clothes pin spring. The lever puts a wheel inside the glass and the spring pushes it down, onto the roller wheels. It is like "fake gravity". I like using this method when engraving several glasses... It is super fast to load the glass... and I don't risk pressing the play-dough too hard (breaking the glass).
3) reduce the accelerations, rapid travel speeds, ..etc.
4) make sure the tail-stock support is not twisted on the rotary base. A twisted tail-stock support will cause the glass to "climb the o-rings" while spinning.
5) Use a 3D printer to customize your gripper and tail stock support to best fit the glass.
6) without a positive grip mount on the rotary device... make sure the project is a single pass engraving design. Don't expect to maintain correct position if your glass is just sitting on o-rings.
I hope we could all help you.. Good luck.