Actually, you're the one falling down a bit of a trap here. Bernoulli and Newton aren't competing explanations at all, and both fully explain the lift (well, you need to use a modified Bernoulli above Mach 0.3 or so thanks to compressibility contributions to flow energy, but that's just a correction, not Bernoulli fundamentally being wrong or anything).

If you take a subsonic wing, look at the airflow around it, and use Bernoulli (or modified Bernoulli at speeds where compressibility becomes relevant), you can fully account for all the lift on the wing. Yes, even if it's upside down. Yes, when if it's symmetrical. Bernoulli doesn't tell you what the flow speed and countries are, it just tells you that if you know the flow speed relative to free stream at some point among a non-dissipative stream tube in the flow, you know the pressure as well. They are linked because all Bernoulli's relation is, fundamentally, is a statement of conservation of energy, and in a flow without significant shock waves and outside of a thin boundary layer, there's no mechanism for much energy loss, so conservation of energy holds.

The Newtonian approach is also true, of course. You'll also fully account for the lift if you look at the downwash behind the wing relative to the incoming flow. This is also based on a conservation law - this is just a restatement of conservation of momentum, while Bernoulli is conservation of energy.

Neither approach actually will tell you the shape and velocity of the flow field though. Newton lets you calculate lift if you know your downwash, and Bernoulli does if you know your flow speeds around the airfoil, but neither lets actually figure out those speeds and angles in the first place. For that, you need substantially more complex math and to start looking at things like flow circulation and the Kutta condition that occurs at a sharp trailing edge.