But because you start with no information, it's very hard to gather more. Suppose Omega reaches into the jar and pulls out a red bead. Does your probability that the second bead will be red go up... down... [or] stay the same...?

My intuition here is to start with an uninformative prior over possible bead-generating mechanisms. (You still have the problem of how to divide up the state space, but that's nothing new.) If a red bead comes out first, I update the probabilities that I assign to each mechanism and proceed from there.

Where exactly that leads seems likely to depend on a bunch of assumptions that I'm too lazy to think through right now (e.g. whether the urn has a finite number of beads in it) but it seems a reasonable way of proceeding in principle.