Bayesians wouldn't assign a specific numeric probability to such an event. Instead they would adopt a prior that was a probability distribution over the probability of the proposition being true. An objectivist Bayesian would probably choose a distribution than encoded only the fact that we don't know what that probability is, and use something like a Beta(1,1) prior which is uniform on the interval [0,1], i.e. every value of the probability the proposition is true is equally likely.
If any evidence came in, they could use Bayes rule to update their prior to produce a posterior, which would also be a probability distribution over the probability that the proposition is true.
Either way, if you want to decide what course of action to take, you work out a loss function, which tells you how much you will lose or gain under each strategy depending on whether the proposition is true or not in reality. We then work out the "expected loss" of each course of action by marginalising over our uncertainty of whether the proposition was true or not (in this case it would involve a sum of the losses, weighted by their posterior probabilities - I'm not going to spell it out on a SE without LateX). We then rationally choose the course of action with the lowest expected loss.
In the case of Pascals wager, the losses consist of:
(i) the cost of behaving like God exists if it does exist
(ii) the cost of behaving like God exists if it does not exist
(iii) the cost of behaving like God does not exist when it does
(iv) the cost of behaving like God does not exist when it doesn't
So the Bayesian scheme doesn't tell you what to do, that depends on the losses, which are going to be your judgement and your prior (which may be an objective uninformative prior). It does however provide a rational means of going from a prior belief and a set of losses (and perhaps some evidence) to the course of action that is most likely to minimise your losses.
For details, the standard reference is Berger "Statistical Decision Theory and Bayesian Analysis", which is published by Springer.
Just a few quotes from the previous question that the OP mentions, first from Keynes:
About these matters there is no scientific basis on which to form
any calculable probability whatever. We simply do not know."
The uniform uninformative prior encodes the knowledge that we simply do not know. It expresses no preference for any probability that God exists (in the case of Pascal's wager). Note it is only an interval in the sense that it covers the entire interval on which probabilities are defined. I suspect "interval" in the previous discussion is likely to mean a subset of [0,1].
And from Taleb:
It eliminates the need for us to understand the probabilities of a
rare event (there are fundamental limits to our knowledge of these);
rather, we can focus on the payoff and benefits of an event if it
takes place.
If we reason from an uninformative prior, and have little or no evidence, the optimal course of action is essentially determined by the "payoff and benefits" (or in this case losses - Bayesians are a pessimistic lot! ;o). The marginalisation over the probability that the proposition is true means that it doesn't affect the outcome greatly simply because it is so vague and uninformative. So it all comes down to the costs. The real problem with Pascal's wager is that the cost of an eternity in Hell is essentially infinite, to all intents and purposes, so that dominates the decision. The difficulty is entirely in justifying the losses.
