It is really hard to see how TR could win the presidency as a third party candidate. (I could certainly see him winning if he got the GOP nomination.) He just did not get enough support from Democrats. Remember, Bryan's showing in 1908 was considered very poor, yet he only lost to Taft by 51.6-43.0.
https://en.wikipedia.org/wiki/United_States_presidential_election,_1908 This means that all a Democrat had to do was hold on to *most* of the core Democratic vote to win, with the normal GOP vote divided between Taft and TR. (For that matter, the Democrats had won control of the House against a *united* Republican Party in 1910!) In fact, I think TR's OTL showing, though far behind Wilson in both popular and electoral votes, was better than it would have been had he not been shot:
"By October, the Bull Moose party showed signs of following the traditional route of American third parties, of proving less potent in November than in August. Three weeks before the election, however, the party received a figurative and almost literal shot in the arm, when a would-be assassin wounded its candidate during a Milwaukee speech. Roosevelt, with his unfailing sense of the dramatic, finished the speech before going off for a two-week stay in the hospital. In an election already decided, his gallantry doubtless reaped a large sympathy vote. 'This shooting will help TR directly by stopping his talking,' assessed Brandeis. 'There seemed to be very strong evidence of an ebbing tide before.' In a probably exaggerated estimate, one Democrat suggesting that the assailant, 'instead of murdering the intrepid Teddy...shot about a million votes into him.'" David Sarasohn, *The Party of Reform: Democrats in the Progressive Era*, p. 148. See my discussion at
https://groups.google.com/d/msg/soc.history.what-if/ZuaG52rwI8c/56xafvBi20QJ
The only thing I can think of that could elect TR would be some terrible last-minute scandal uncovered regarding Wilson or whoever the Democrats nominated. But if TR won because of that, his victory would probably be considered a fluke, and Democrats and Republicans in Congress would see little need to change their party affiliations. (Presumably there would be more than the 13 congressmen elected as Progressives in OTL, but probably not very many more. Observers at the time noted that the Progressive vote had "an 'inverted pyramid aspect.' It is largest at the top and 'tapers down very fast.'"
https://books.google.com/books?id=FJ5FAQAAMAAJ&pg=PA997 This is not *totally* fair--two Progressive candidates for governor, Albert J. Beveridge in Indiana and Oscar Straus in New York, won more votes than TR in their states--but in general the Progressives did lag behind TR in down-ballot races.)
