No genuine conservative had any chance to get the Democratic nomination in 1912. If it isn't Wilson, it will probably be Clark, who, despite Bryan's grumbling, was an effective and progressive legislator.
The two "conservatives" most often mentioned--Judson Harmon and Oscar W. Underwood--were vehemently opposed by Bryan, and had very little chance of getting the nomination. Yet even they were not reactionaries. (No doubt Bryan had a grudge against Harmon for his failure to support Bryan in 1896, when Harmon was Cleveland's Attorney General. This was also true of Wilson, of course, but to Bryan the cases were different: one was "a trained officer in the Democratic army," the other "a scholarly recluse."
https://books.google.com/books?id=wKAeAQAAMAAJ&pg=PA66) On Harmon's record in Ohio, David Sarasohn writes in
The Party of Reform: Democrats in the Progressive Era, pp. 112-13:
"Just as La Follette's Wisconsin served as the insurgent ideal, Ohio became the Democratic reform showcase. Governor Judson Harmon, despite Bryan's constant attacks on him as a reactionary, signed a sizable amount of progressive legislation. Johnsonian Democrats [i.e., followers of Tom L. Johnson, reform mayor of Cleveland] in the legislature may have fought harder for the bills than did the management-minded governor, but Harmon supported their efforts, and appeared before the party caucus to remind it of its obligations to the platform. In his first term Ohio reorganized its tax structure, raising the valuation of railroad property from $166 million to $580 million. In his second term, a more Democratic legislature set up a Public Utilities Commission and passed a strong corrupt practices act, a ten-hour maximum workday and a fifty-four-hour workweek for women workers, and an optional worker's compensation bill. The latter, and other labor legislation, was introduced by William Green, a Democratic state senator from the coal districts who would later succeed Gompers as the head of the AFL. 'It would be unfair to Judson Harmon,' argued his successor [James Cox], 'to assert that he was not a constructive liberal.'"
As for Underwood, Sarasohn writes (p. 99) that he "had come to Congress in 1895 as a strong supporter of labor, free silver, and an income tax and rapidly rose to leadership positions. If his later marriage to a Birmingham steel heiress seemed to have taken the edge off his reformism, he was still far from a standpatter and closely followed the party line on most issues." In any event, Underwood was almost purely a regional candidate, with very little support outside the South.