It is only either/or in this thread, not in real life.
Well then the original point is moot then if it does not exist in real life

A man can't force himself to be physically attractive to someone he is not, nor do men necessarily have to trade in mental stability to have someone who cares for their outside appearance.
I would guess the choice is more like this when you don't know either person really well on an initial meeting: "with all things being equal on the inside from what I can tell, I'll take the physically attractive one".
As much as us girls would like to deny it, men are visual creatures. I think that initial attraction means a lot because of that and when you consider most people do not know each other well on that initial attractiveness meeting, then it should answer your question of what men would pick and why...at least initially. Most would probably ditch the headcase though no matter how beautiful if given the chance AND THEN look for another attractive woman that has ALL the qualities that they want. Just a learning and growing experience for most.
Plus I would submit that some people would think a very overweight person is wearing their mental health issues on the outside already kinda like one would be saying get away with 20 facial piercings. Kinda like a big warning flag. It may not be true necessarily, but it probably does give someone pause if you don't know the whys of why the weight is there becuase there are plenty of women for whom the weight is intimately tied to mental health issues they have. The argument obviously being that with a slender woman, she's not wearing a badge of her issues right on her sleeve for all the world to see which makes her more agreeable, initially, to your average guy.
I wasn't trying to ruffle your feathers, mama, I just think you've given the guys an impossible and unrealistic choice. I think you have a great thread going here, offshoots and all.

Heck, I've been insanely busy and I got out for this one and the ewww thread so
