No violation of Bernouilli required.
If an aircraft had a symmetric section wing (as do some aerobatic planes, for example) then in order to generate some lift it needs to fly at an angle of attack, which essentially makes the wing non-symmetric relkative to the airflow, even though it is geometrically symmetric.
If the same aircraft is flying 'upside down' and wants to generate the same lift in the opposite direction (which is the "new up") it simply flies with it's wing at the same but opposite angle relative to the wind, so that the wing is essentially turned upside down and still generates the same lift in the same way.
For most aircraft, the wing is shaped - cambered - so as to be most efficient generating lift in the traditional direction. But all that means is that you need to have a greater angle when upside down, and it's not as efficient. It's still possible.
If wings could not generate "lift downwards" relative to the aircraft, outside loops and "negative 'g' " manoeuvres would be impossible. They aren't.