If TR somehow won election as a Progressive, Democrats and Republicans in Congress might unite on a presidential-term-limits amendment. But I think such an election is extremely improbable--almost any plausible Democratic presidential candidate would win the core Democratic vote, which would be enough for victory in a three-way race. IMO the only plausible way for TR to be elected in 1912 is as a Republican. If that happens, the Republicans will oppose such an amendment--at least as long as TR is alive.
After TR's death, a Democratic Congress might pass such an amendment as a posthumous slap at TR--the way the 22nd Amendment in OTL was a posthumous slap at FDR by the Republican 80th Congress. And just as the 22nd Amendment got some support from Democrats who had not cared all that much for FDR and felt freer to implicitly criticize him now that he was no longer around, so this amendment will get some support from Republicans who (even if only privately) had never cared that much for TR.
On second thought, the amendment that a Democratic Congress might pass after TR's death might be quite different from the 22nd Amendment--it might instead be a single-six-year-term amendment. Bryan had favored this; it was in the 1912 Democratic platform [1]; and in OTL it was only defeated through Wilson's efforts. [2] But in this ATL Wilson is not elected president in 1912--and probably not thereafter, either--so Bryan might get the Democrats to go along with this pet idea of his.
[1] Although actually the platform doesn't even specify that the term be six years: "We favor a single Presidential term, and to that end urge the adoption of an amendment to the Constitution making the President of the United States ineligible to reelection, and we pledge the candidates of this Convention to this principle."
https://www.presidency.ucsb.edu/documents/1912-democratic-party-platform
[2] See Wilson's letter to A. Mitchell Palmer at
https://www.alternatehistory.com/fo...-year-presidential-term.395351/#post-12902607 As
The Nation remarked when the text of the letter was made public in early 1916, "From any point of view, it is a noteworthy document. Nowhere in it did the President even allude to the Baltimore platform, favoring a single term, urging a Constitutional amendment to make the President ineligible for reelection, and pledging 'the candidate of this Convention to this principle.' The candidate of that Convention never made any public reference to this particular plank; and his letter to Mr. Palmer, written in February of 1913, showed that he regarded it as of no binding force. He wrote, in fact, as if it had never existed. It cannot be said, therefore, that he has changed his mind in order to suit his present political ambitions. From the first he treated the one-term plank as a bit of Bryanesque buncombe."
https://archive.org/stream/nation102jannewy/nation102jannewy_djvu.txt