My final solution was to just use a drill press to drill a 2" hole with a forstner bit partway through a block of wood, then a 5/16" hole the rest of the way, centered on the pilot hole from the forstner bit:
Then I just stick it on the end of the dowel and use it as a drill guide:
This worked well. It's not really sensitive to imperfections in the roundness, and since I have to go to a friend's shop to use the drill press, I only had to use the press once and now I can use my handheld drill at home for drilling the dowels.
Prior to making the jig, I was at least able to find the center using an Incra IG32, by setting it to 1" (the radius):
Marking a line down the center:
Then moving it and repeating:
I made a third line just to verify (I was worried about the center notch in the gauge but the effect was negligible).
Still need to drill the hole precisely but that's at least one working solution for marking. Any graduated marking gauge that you can set to the radius will do.
If you have two framing squares you could clamp them together in opposite directions to make a 1" x 1" (or whatever radius) square opening, line one up with the dowel edges, and the inside corner of the other will theoretically be the center, as long as you clamped your squares precisely. This should also work for finding the center of a circle drawn on something flat, unlike the dowel which is an actual, physical circle.