No, the dilemma depends on having many copies. You're trying to optimize the outcome averaged over all copies (before the copies are made), because you don't know which copy "you" will "be".

In the no-copies / amnesia version, the updateless approach is clearly correct. You have no data to update on - awakening in a green room tells you nothing about the coin tosses because either way you'd wake up in a green room at least once (and you forget about it, so you don't know how many times it happened). Therefore you will always refuse to play.