I thought you might like to hear about some of the literature on this problem. Forgive me if you're already aware of this work and I've misunderstood you.

Manfred writes:

If people are memory-wiped at some interval, then this increases the probability I should assign to being in room B - probability of being in a specific room, given that your state of information is that you suddenly find yourself in a room, is proportional to the number of times "I have suddenly found myself in a room" is somebody's state of information.

In Anthropic Bias: Observation Selection Effects in Science and Philosophy, Nick Bostrom describes a thought experiment known as 'Mr. Amnesiac' to illustrate the desirability of a theory of observation selection effects that takes this kind of temporal uncertainty into account:

Mr. Amnesiac, the only observer ever to exist, is created in Room 1, where he stays for two hours. He is then transported into Room 2, where he spends one hour, whereupon he is terminated. His severe amnesia renders him incapable of retaining memories for any significant period of time. The details about the experimental situation he is in, however, are explained on posters in both rooms; so he is always aware of the relevant non-indexical features of his world.

Not unlike Manfred's arguments in favor of betting on room B under imperfect recall, Bostrom's solution here is to propose observer-moments, time intervals of observers' experiences of arbitrary length, and reason as though you are a randomly selected observer-moment from your reference class, as opposed to just a randomly selected observer (in philosophy, Strong Self-Sampling Assumption vs. Self-Sampling Assumption). With this assumption and imperfect recall, you would conclude in Mr. Amnesiac that the probability of your being in Room 1 = 2/3 and of being in Room 2 = 1/3, and that you should bet on Room 1.

But I don't think there's anything mysterious there. If I understand correctly, we are surreptitiously asking the room B people to bet 1000 more times per observer than the room A people. Yet again, the relevant consideration is "How many times is this experience occurring?"

Nitpick: If we do include imperfect recall, doesn't this actually just make us indifferent between room A and room B, as opposed to making us prefer room B? Room A people collectively possess 100 trillion observer-moments that belong to 100 trillion observers, room B people collectively possess 1000 observer-moments per observer times 100 billion observers = 100 trillion observer-moments that belong to 100 billion observers. Our credence should be 50/50 and we're indifferent between bets. Or am I confused?