How can I create a center disk to put under a compass point?
For something of this sort, you'll either construct one yourself, or repurpose a common item from another resource. The first thought that popped into my alleged mind is a coaster, something with a reasonably solid bottom layer to prevent puncture and also to provide grip, and an upper layer with material suitable to hold the compass point.
Even a sheet of hobby plywood cut in a shape to meet your needs should work. You'd be able to cut more than one, to have replacements when the point holes are excessive in quantity.
Shelf liner and drawer liner made of an open mesh material that's spongy is found in grocery and big-box stores and can be applied to the bottom of the coaster or coaster substitute to prevent slipping.