Repeatedly cutting 44mm circles from paper
If you can provide links and/or images, my answer can be adjusted, but at first thought, consider the following.
For compass style cutters:
The pivot of the cutter could be attached to a metal disk or to a magnet. The size and shape of the disk or magnet would be dependent on the structure of the pivot.
If the pivot is a magnet, the underlying surface should be metal, but you'd want a cutting mat between the paper and the metal sheet.
If the pivot is a metal disk, the magnet should be embedded in a wood surface or if the magnet is thin enough, cut a matching recess in the cutting mat or equivalent.