You can't cut very deep with a 1/8" router bit without going painfully slow. It would be faster by far with a Scroll saw if you have the right blade. You should be able to cut 1/4" pieces pretty fast with the correct blade.
I was experimenting with using a metal working parting tool that is not a deep as woodworking parting tools. It would let you get into the tighter rings. That got put on the back burner when I moved and I just found it the other day. Don't know when I will have time to experiment with it some more.