You can also use a bearing race with a large ball bearing, available for the picking up at very large equipment repair shops - or if you know someone who works in one. A ball 1 1/2" to 1 3/4" works well for most patches, even for the Bess.
Bearing races are also easy to come by - they don't have to be new.
We've been using this system for many years - many of us and they work well. NEVER had a race or bearing break, BTW - you only cut one at a time, but with thin material some guys use, ie: up to .015", you could probably cut 2 at a time. It only takes a few minutes to cut 100 patches.
Guess a picture or two would be a good idea. here are a variety of punches- on the left, arch punches, 2 commercial and one home made, welded up one for .36 and .38 cals. The races are a variety of sizes and include one large diameter U-joint cup which also works. You can see the cloth patch, ready for cutting- with a smack of a hammer- but not on the counter. I use the anvil of the bench vice for backing, while Taylor used an anvil attached to a 'round' of Pine or Spruce. The backing should be solid. AND - NO - None of us had ever split a ball or bearing race- no flying metal chips- will not happen for those overly concerned with safety. Do wear glasses - just in case, but I repeat, none of us has ever had a bearing break - it doesn't take much of a smack with a hammer on the ball to cut a perfect patch, every time. We've cut thousands and thousands of patches this way. The Arch punches work best with a backing of UHMW (HDPE - USA). I picked up a couple at a specialty plastics shot where they make them for chain bearings for lumber yards, benches for butcher shops, etc. Scrap 2" thick piece 6" X 12" cost me 10 bucks. It will never wear out - well, not for a long time, anyway. I've already cut a thousand patches for the .69 using the large 1 1/2" arch punch on the left. The next smaller one is 1 1/8", OK for .54's and .58's. The next one is for the .36's to .38's. It's a bit small for the .40. The larger balls int he picture are 1 5/8".
I should add that I sharped the ege of a new race by holding the bearing race hole against the 8" grinding wheel. Light pressure works just fine and the race spins itself as it sharpens the edge, making an even grind. It needs touching up periodically, maybe every 500 to 1000 patches due to slight peening. A race and a ball are easy packing for going on a trip to a rondy just in case you run out of pre-cuts. Depending on the thickness of patching, you can cut from 8 to 20 patches before having to stop and push them out of the race onto a pile. Taylor cuts his cloth strips 5 or 10 patches wide, so he can make piles of 10 to keep track of the number cut. It takes very little time to cut 100 patches as I noted.