The lift can be calculated by integration of the pressure over the surface (and than subtracting buoyancy). But in order go get the pressure profile according Bernoulli the velocity profile must be given. If you are looking for a general proof, not only for a special case with known velocity profile, then Bernoulli alone is not sufficient. I'm not even sure if such a proof is possible because we don't have general solutions of the Navier-Stokes equations. But we can check it for every special case. Do you have an example for a velocity profile that doesn't match the pressure profile according Bernoulli?
Why should that be a problem? Lift results from the velocity profile over the whole surface of the airfoil and not from the velocities at two special points only and with different velocities you get different forces even if top and bottom side of the profile have the same length.
No, I don't have example velocity profiles. I am just going by statements in lectures and technical notes that I have come across over the years. The nature of this is that I do not have references either. I am not insisting anything here, it just seems to me a point of some controversy that I’d like to better understand. In truth, I probably don’t have a high enough technical background to do so but why not ask and see if I can learn something at least.
“Why should that be a problem?...” Well, that whole concept of upper surface greater distance in the same time, just always seemed too schoolboy simple and convenient. So, I’m fishing to for enlightening comment.