What kind of proof would it take for people to agree that the obesity epidemic is due to food being dramatically cheaper and more palatable than in our environment of evolutionary adaptedness?
Did you read the linked paper? They consider this possibility and reject it.
Sure. Did you realize that epistemically they don't have the power to reject explanations? It would certainly be news to Stephan Guyenet if they did have in this case.
How would that explain lab animals gaining weight?
The most likely explanation for this is some combination of 1) lab animals being fed more over time because researchers themsevles have been normalized to bigger food portions, 2) incomparable data of lab animals feeding over time, and 3) fraud because of the pressure to show unusual results.
The theory calories-in-calories-out doesn't explain why animals fed controlled diets over time could exhibit a change in weight, pretty much by definition. So instead of trying to extend that paradigm in some way that would explain it -- or even to acknowledge that there could be some auxiliary cause for this phenomenon while saying it's irrelevant to the trends in human obesity, which I think would be a fair argument -- you're saying 1) they aren't actually being fed the same thing, 2) they aren't actually being fed the same thing, 3) they aren't actually being fed the same thing or they aren't actually gaining weight. This is only "most likely" if you begin by assuming the truth of what's to be proved, i.e., begging the question.