If there is any kind of a market for these, I'd suggest you farm it out. Making your own will be extremely costly per unit. If you use ply, it will have to be Baltic Birch because the layers will show and anything less will not look very good because of gaps, etc.
They could be made from glued up stock. Buying up cutoffs and scrap is one solution, but you'll need to joint and plane them for uniform thickness before gluing them up, but that would allow you to leave the centers open so you have less waste. The look would be even more interesting than ply, and that could be a sales feature. The glue joints will hold forever and it will take an interesting finish. You could name them something like "Eco-rings" to stress that they are made of recycled material and are Earth friendly, that could justify a little higher price.
I guess it depends on what size enterprise you're planning. If you are very connected to the gymnastics community, you could make this a premium item. I always go upscale.