If possible, I like to fill gaps with sawdust from the adjacent wood. On small checks and cracks John Lucas taught me a trick: wet sand with thin CA glue on coarse sandpaper. But that’s a pretty big gap. Do you know if it gets larger or smaller as more wood is removed?
I wonder if it’s still concentric while in the cole jaws if it might be possible to a thin, even gap all the way around, then fill it with something dark or glue in thin strips of veneer.
BTW, gluing up layers can be tricky. I like to run the layers through the drum sander to get both surfaces parallel. There’s a better method that doesn’t use a drum sander but involves turning and truing rings one at a time but it’s more involved. If I can find the document I use for teaching multi-layer boxes, I’ll try to post it. (Unfortunately, I may not be able to get to those files for a few days.)
Oh, I just found a file with five pages of “unpolished” drawings describing the multi-ring method. I’ve used this method to make a number of multi-layer “Beads of Courage” boxes. It works extremely well and always had tight joints. From what I can see of your design, the middle layer would need to be glued up separately. A drum sander, or better, a jointer would be great for this if you have access to them.
JKJ