How should I secure the inner ring of bearing race to a wooden frame?
Since wood glues tend to be stronger than wood itself in my experience anyway, gluing the dowel to the board sounds like a solid plan to me with one exception...

I'd suggest reversing the order.  Secure the dowel in the bearing first, since that step will put more potential strain on both parts.  Particularly if the dowel breaks, you don't want it to be permanently affixed to the board when it does.

Compared to the force necessary to wedge the dowel into the bearing, the insertion into a hole in the board with glue in it will be trivial amounts of pressure.