Teddy was a brown belt in judo and ju-jitsu, Lincoln was a celebrated catch wrestler. Who wins in an mma bout?
Teddy has the boxing experience, Lincoln the reach....are we talking prime-for-prime or during their time in office? Lincoln is a slight favorite here. He has a championship of sorts in his past, and his lack of striking experience will be offset by his superior abilities inside and greater height. Jiujitsu will not be enough to save Teddy, especially if it's the variety he supposedly learned from John O'Brien. Lincoln by submission...unless he practiced a variety of catch wrestling that does not include submissions (plausible, actually--unless it's the Lancashire variety, it would only have "falls"). If that is the case, then Teddy would either submit or KO Lincoln. The latter is more likely.
I'm picking Roosevelt, Lincoln was skinny, Roosevelt most likely hits harder and he's just the kind of person that when he hits, he wants to take your head off, whereas Lincoln seems more of a gentleman fighter.
Teddy can bang it out, and I think he can use his stout frame and judo experience to stay on his feet. Lincoln's no-gi grappling would give him an edge on the mat, with his reach he could ground and pound teddy easy from the gaurd. It's a tough fight. Probably the two presidents who would do best in fighting, no?
Bush clobbers Lincoln with a well-thrown baseball, only to be killed in a hail of bullets from T.R.'s elephant gun.