A #22 cutter, for making upholstery buttons, is 1-1/16" in diameter.
It can cut through 15-20 layers of upholstery fabric in a snap. In
the link, the cutter is the center piece and can be purchased
separately for about $30. Just about any upholstery supplier in your
area will have one in stock. It is hollow, so you would still need to
punch the cut material from the center. When cutting fabric manually,
a Handy button machine is used to press the cutter through the fabric,
but a mallet can be used, also.... just keep the cutter straight/
vertical onto the surface.
This cutter could probably be mounted (chucked) on some sort of press
machine of your own making, for mass cutting. I don't think the issue
of removing the cut disks, from its center, would be difficult, at
all. I think if you saw one of these cutters, an easy mounting/
chucking technique could be readily devised for your mass production
When cutting fabric with the manual Handy button machine (metal base),
a backer board (wood) is used to prevent the cutting edge from
becoming dull or chipped, on the metal base. Furniture companies use
pneumatic Handy button machines for cutting fabric disks and making
buttons enmasse, so the option of getting one of these pneumatic
machines is there, also, if you are to do this long term.