The likelihood of encountering the winning side of the bet is proportional to the likelihood of encountering its losing side. As such, whether you are likely to encounter the bet once in your lifetime, or to encounter it a hundred times, doesn't seem to significantly affect the decision theory you ought possess in advance if you want to maximize your utility.

In addition to Omega asking you to give him 100$ because the coin came up tails, also imagine Omega coming to you and saying "Here's 100,000$, because the coin came up heads and you're the type of person that would have given me 100$ if it had come up tails."

That scenario makes it obvious to me that being the person that would give Omega 100$ if it had come up heads is the winning type of person...