The crux of the matter:
I would rather you cooked tonight. (BrE) = I would rather you cook tonight. (AmE)
Both forms mean the same. It's about the future action.
I would rather you were cooking while I am reading now. (BrE) = I would rather you be cooking while I am reading now. (AmE)
Both forms mean the same. It's about the on-going action in the present
I would rather you had cooked yesterday. (BrE) = I would rather you have cooked yesterday. (AmE)
Both forms mean the same. It's about the action in the past.
Do you agree that these are legit and grammatically correct? Do you use them?
I would rather you cook tonight. (AmE)
I would rather you be cooking while I am reading now.
I would rather you have cooked yesterday. (AmE)