While I am a fan of FNaF, I feel that Rammus would be more likely to win. I'm saying that because given the way Dreadbear looks, I feel that it's pretty easy for Rammus to hit Dreadbear's weak points. Not to mention that I really don't have to have a clue how League of Legends works to know that Rammus' spikes would make Dreadbear's blunt force attacks wind up hurting Dreadbear instead, especially after a Defense Curl. Seriously, try stepping on a sea urchin; it'll ******* hurt.
Also, bear in mind that in FNaF 3, it was shown that animatronics will die by being taken apart, so it's really a matter of whether Rammus could take Dreadbear apart. More than likely, I think Rammus would get the win from Dreadbear mainly due to Dreadbear's faults rather than Rammus' skill advantages.