It should be possible to efficiently compute the deterministic behavior of an infinite population under these rules. In particular, the presence of a slow strategy should not make it hard to simulate a large population for lots of generations. (Assuming strategies get to depend only on their current opponent, not on population statistics.)

First find the expectation of the scores for an iterated match between each pair of strategies: If they're both deterministic, that's easy. If there's some randomness but both compress their dependence on history down to a small number of states, then dynamic programming works. And in the general case, approximate it with an empirical mean.

The aforementioned expectations form something similar to a Markov transition matrix. Then updating the infinite population by 1 generation costs only O(N^2) multiplies, with N being the number of different strategies.