Had he been elected in 1912, I doubt Roosevelt's health would have carried him to Versailles. Even if it were an earlier Versailles Conference in 1918. Many credit TR's death in 1919 to the diseases he acquired during his 1914 trip to South America. That's only partly true. His death was due to a lifetime of overstress, war injuries, adventurism, and the death of his son that devastated him in 1918. Roosevelt historians pretty consistently point to his son's death as the largest and most direct contributing factor to his fatal heart attack. Given that life expentancies for pilots during WWI were extremely low, if Quentin follows a similar military career following his father's early entry into WWI he is almost certain to die in roughly the same amount of time as OTL. (Just over a year after the US declares war). This plus the general stress of wartime leadership means that if elected in 1912 and reelected in 1916, Roosevelt will barely make it into his second term before dying at some point probably in 1917 or 1918. (In OTL, he nearly died in February 1918 due to a rectal abscess. I think that had he made a Presidential comeback and served for another five years since 1913, this health incident would have killed him).
That said, also contrary to popular belief the League of Nations was not Wilson's brainchild. It was a policy he pushed for, but an international peace keeping organization had been talked about in the US and UK long before Wilson's 14 Points. Roosevelt and Taft publically advocated for such an idea, which also had the strong support of Lloyd-George in Britain. Some sort of international body would have been created by the Allies after WWI, if not America it would have been Britain leading the charge. In fact had Wilson not been President to shoot down US involvement because he was too arrogant to compromise with Lodge, America might actually have a role in this new organization.