Use of a flex coupling like you depicted will require that the driven shaft have well supported bearings. That type of coupling is used to compensate for misaligned shafts. Depending on your design, a rigid coupling might serve you better.
Based on your comment above, I would assume that you don't have a well equipped machine shop. More than likely, a lathe would be required t make a "close fitting hub or coupling as the likelihood of the poly-V pulley being a diameter that would fit standard bore diameters is small.
Also, my understanding of how a tumbler works, the rotation has to be a low enough rpm that the tumbling media can fall due to gravity, thus mixing the media a parts (vis. a front load washing machine or cement mixer). Too fast and the centrifugal force will just keep the media pinned to the outside of the drum. (One can calculate the maximum rpm fairly easily, depending on your drum diameter.) My guess is that you will probably want something like 50 rpm. Most treadmill motors have a rated speed of around 4,000 rpm which would be an 80 to 1 reduction. As you surmised, this may require some means of mechanical speed reduction in order to achieve your desired speed.
A small gear box would be my choice as it is difficult to obtain large speed reductions with a single pulley arraignment. I would suggest maybe the front end of an old drill or similar. Drill or drill/driver motors operate at similar speeds and are geared down to usable speeds at the chuck.