I think OP should go with the better PI, however suggestion Stanford and Berkeley have the same level of prestige seems wrong to me
Outside of the physics community, there may indeed be some differences in reputation, though I'd expect them to be small, since anybody hiring physics PhD's will have a rough knowledge of Berkeley's excellent reputation in physics. Sure, in other fields (or for undergrad), the reputations may not be equivalent. NB: I'm Stanford undergrad, MIT physics PhD, so I have no personal reason to hype up Berkeley (the opposite bias, in fact...).
Some data: Last I knew, Times Higher Education surveyed published tenured professors worldwide about the best schools in their field. I think it's probably the best and mabye only good data.
Every year, six schools have been head and shoulders above the rest overall (in random order): Harvard, MIT, Oxford, Cambridge, Stanford, and Berkeley.