I do not think it is missing - since a Bayesian can ask themselves at what odds they would accept a bet on the coin coming up heads - just as easily as any other agent can.

What is missing is an account involving Bayesian forgetting. It's missing because that is a way of solving the problem which makes little practical sense.

Now, it might be an interesting exercise to explore the rules of Bayesian forgetting - but I don't think it can be claimed that that is needed to solve this problem - even from a Bayesian perspective. Bayesians have more tools available to them than just Bayes' Law.

FWIW, Bayesian forgetting looks somewhat managable. Bayes' Law is a reversible calculation - so you can just un-apply it.