They Republicans weren't in that much trouble. Taft + Roosevelt together beat Wilson by 10 points and beat Wilson in Missouri. Throw in a complacent candidate that his opposition can define to the public as more conservative than he really is and you have something to work with.
Was Wilson's image any less conservative? Frex, in the 1912 campaign he never came out for women's suffrage, which TR did and which Bryan had done before him. Clark, OTOH, had made the first moves to organise Democratic women nationally, and his sister, Annie Pitzer of Colorado, was one of the two women delegates at the Democratic Convention that year. Wilson also opposed a national child labour law and minimum wage legislation, which TR supported.
If Clark has been "painted" as more conservative than he really is, he has the entire campaign in which to repair any damage. William Randolph Hearst was a strong Clark supporter, so his papers will probably endorse Clark more enthusiastically than they did Wilson. Might he even be Clark's running-mate? And Bryan, whatever problems he might have with Clark at the Convention, is a passionately loyal Democrat and will certainly campaign for Clark in the general election, as he did for Parker eight years before, though the latter advocated policies far more repellant to him than Clark is likely to do.
I can't help feeling that the widespread notion of a "conservative" Clark is largely a product of hindsight. We all know (if we've studied the period at all) that Wilson, once elected, turned out to be a really strong progressive, pushing through a whole shaft of reforms, so that's how we imagine him campaigning. In reality, though, he ran a pretty standard Democratic campaign of that era, certainly sounding less radical than Bryan normally had. There is no reason to think that he came over to the voters as any more progressive than Clark would have done.
Finally, even if Clark
did cultivate a (slightly) more conservative image than Wilson, is there any real evidence that it would have harmed him? If he lost some "Wilson" voters to TR, he would probably make some gains from Taft, leaving the overall effect pretty much "swings and roundabouts". I certainly don't see the slightest reason to expect the massive losses that your scenario requires.
This is not to say that nominating Clark makes no difference, but my guess is that the effects are more geographical than ideological. Wilson, an easterner and former President of an Ivy League college, probably did better in the northeast than Clark would have. I can imagine Clark losing Maine to TR and the rest of New England to Taft, while TR probably increases his lead in Pennsylvania and has at least a chance to pick up New Jersey. Further west, though, Clark should do about as well as Wilson, and maybe better. In the Midwest there is probably no change (TR had fairly solid majorities in the three states he carried there) except that Clark, as a next door neighbour, may take Illinois by a slightly bigger margin than Wilson did. However, if Clark comes out for women's suffrage (quite possible, given his record) he should do better than Wilson in those western states where women already vote, so almost certainly winning California, which Wilson lost by a razor-thin 0.025% (that is
not a misprint) and has a slimmer chance of edging out Taft in Utah. That gives him a comfortable 392, against 97 (including Maine and NJ but excluding California) for Roosevelt and 42 (still including Utah) for Taft. I don't expect the popular vote to change much from OTL, though it will be differently distributed.