A student shows up in a class that is not theirs — how do I record it?

Through the make-up route, not by adding them to that class's roster. A student can only be marked in a class they are enrolled in, because that is where the pack that deducts the session lives.

If the move is permanent, the action is not attendance but placement: the student is placed into the new class with a payment model, and the old enrollment is ended. Attendance history in the old class stays intact.