I backed off from trying to count n-simplices with all their faces and hyperfaces oriented in a composition-consistent way, and instead tried to count the number of distinct n-simplices with just the 2-cells oriented.
Because the category Δ category theorists use to define simplicial sets is just the category of ordered sets, i.e. the category of simplices with all the 1-cells oriented. I'm just trying to move up a dimension! But already it seems really hard to count the number of possible 2-orientations.