I stopped using metal rings for my medicine wheels and dreamcatchers. With high humidity, I had a few that rusted and it came through the leather wrapping. I've been using willow (for baskets) when I can find it, or the wild grape vine that's everywhere around here.
Both can be soaked to make them more pliable. Using a piece with a larger diameter (1/2 inch or larger) I make mitered cuts on the ends for a smooth joint. With smaller pieces, I overlap and secure with synthetic sinew and hide the joint with my leather loop for hanging.
With the grapevine, it only takes 1 or 2 pieces. I start with a piece at least twice the length of the desired circle size. Make the circle and then wrap around that, tucking the loose ends between the wraps.
Both of these look great wrapped in scrap leather or plain. At your home, humidity probably isn't an issue with the metal rings. But then you know me, traditionalist at heart. Rust wasn't my only issue for finding a natural alternative.