Your PS, in which you say to "assume that in the overwhelming measure of the MWI worlds it gives the same outcome", contradicts the assertion that the coin is fair, and so you have asked us for an answer to an incoherent question.

This doesn't sound right to me. The coin doesn't need to be quantum mechanical to be fair. Here is a fair but perfectly deterministic coin: the 1098374928th digit of pi, mod 2. I have no idea whether it's a zero or one. I could figure it out if you gave me enough time, as could Omega. If both of us agree not to take the time to figure it out in advance, we can use it as a fair coin. But in all Everett branches, it comes out the same way.