Under these circumstances, I think TR would endorse him and stay out of the race. I really doubt he would be nominated a 4th time in 5 elections after losing the first 3 times, though. Charles Bryan would be more plausible if he got into politics a bit earlier in his life.
Endorse Taft or Bryan? If the latter, why would TR endorse Bryan?
"They were both progressive!" After all, it's not as though Theodore Roosevelt was a lifelong Republican and that he returned to the GOP fold after 1912, nor is it that in 1896 he made a job of hammering Bryan as the scourge of progress and merit. Moreover, Roosevelt likely deplored Bryan's view of America's place on the world stage, but disregard all of that because a New York aristocrat would cast his lot with an agrarian anti-imperialist owing to the fact that an eleventh-grade history textbook might label both as progressive.