Can you explain how an OAK above the stove would be a problem and how much of a difference you think is passive air opening vs OAK?
If OAK brings in cold air wouldn't that make the chimney pipe colder when the coals go down?
An OAK with an inlet above the stove may function as a chimney (and thus while burning creates a fire hazard - hence this is forbidden by code, and this is a major problem).
Regarding your issue: Having 2 "chimneys" compete for air flow direction might very well result in the heavy stack of cold air in the real flue pushing down and winning - i.e. reverse draft. Opening the door at that time (when this flow from the top of the chimney to the OAK inlet, rather than the reverse, is occurring) would allow this big flue of cold air to drop down in your basement as streaming out of the door has a lower impedance in your (low pressure) basement than thru the stove and the narrow OAK out again.
When you (before starting the stove) notice reverse draft, close the stove door (leave the air setting as it was), go outside, light a cigarette or incense or something near the inlet and see how the air flows at the OAK inlet: outward (reversed) or into the OAK (normal).
OAKs are good for tight homes where make-up air is hard to get. But OAKs can become very dangerous too.