Fuzz the turn on which Luck times out by hashing ubirthday together with (turns/300), and taking the result modulo 300. This means that Luck will still time out once every 300 turns, but the player can’t predict the exact turn on which it happens, and figuring out the exact turn it happens in one iteration is unhelpful for any future iterations.

If Luck is timing out once per 600 turns, still do the above, but do nothing on every other iteration. Determining which set of “every other” iterations to use could also be done by hashing ubirthday and seeing if the result is odd or even.