He's very look like a lion, act like a fox.

Oh, cool, yeah! That's another good way to think about it, because you're completely right about this:

Everyone looks at his war hero reputation and conveniently forgets that he won those wildly overmatched battles through cunning rather than strength,

Cunning, and also also a willingness to manipulate / go against convention -- very Slyth. I hadn't thought about that bit of history, but absolutely, that's strong evidence of his Slytherinness.