Who is correct about the sun exploding actually is irrelevant to that; only the conditional probability of the bet being collectable if the sun has exploded vs. that of it has not exploded is needed here. You would care about the probability that the sun actually had exploded if one of those weren't zero, but it is, so it doesn't matter.
The Bayesian is correct to offer the bet.
Who is correct about the sun exploding actually is irrelevant to that; only the conditional probability of the bet being collectable if the sun has exploded vs. that of it has not exploded is needed here. You would care about the probability that the sun actually had exploded if one of those weren't zero, but it is, so it doesn't matter.