Hmm. 'Tis an interesting question.
It boils down to: can American politics adapt to having a third party?
The Democrats, of the time, are a national party with a racial veto by Southerners and in Presidential elections are mostly confined to the South. They are simply not competitive on the presidential level, although their Southern lock usually gives them control of Congress in part or in full.
The Republicans, then, are a non-national party confined to all areas outside the South. They have to win 2/3 of all non-South electoral votes… but they almost always do so. Five phases after the Civil War: pre Teddy Roosevelt; Teddy, post Teddy, post Great Depression, post Goldwater.
Before Roosevelt they won because the Democrats were tied to the unpopular South, because the public didn't care about their big business bias, and because the conservatives and liberals agreed more with each other than the Democrats
Just when their conservative pro-business over people slant was about to hurt them big-time (if, for example, Bryan had held different issue positions) they got lucky with Teddy.
Teddy was popular partially because of his populist reformer tinge, taking on big business and so the less ideological reformer elements flocked to him and the conservatives had nowhere else to go. I, and others, tend to group the reformers with the liberals but that's mostly because the various conservative wings tend to drown out the differences of the other side.
Post Teddy the progressives were pissed and marginalized, but like the conservatives had nowhere else to go after 1912 and so just came back to the GOP every four years and post Great Depression after a couple of conservative Presidents looked really bad (phase four) they basically forced moderate to liberal Presidential nominees.
All the way until the conservatives took over in '64, and left the reformers with nothing; the liberals and moderates with a shrinking base going to the Democrats.
(See Ross Perot and Barack Obama for modern reformer led candidacies. Less ideological, more fix the system. Reformers used to be in the GOP, then they got shoved out after Goldwater and were part of Perot and now they seem to be part of the Democratic Party.
So, really, progressives today are just avoiding the term liberal. Actual progressives go way back, are less ideological more "whatever works", and are basically reformers. They were part of the GOP, lost influence after Goldwater, came back ('94, Newt, along with Southern demographics), and now seem to be Democrats or swing voters.)
The key, for the Progressive Party, would be to consolidate support. Leave a third of the country to the GOP conservatives. Leave the South to the Democrats. Focus on your reformer base along the North of the country, and the attainable moderate/liberal base in the Northeast.
Under that I imagine the Progressives could muscle out the GOP and Democrats in a lot of House seats, and some Senate seats. The problem arises in Presidential races. As in 1912 if the Republicans and the Progressives split the non-Southern vote three ways—the Democrats win.
Possibly the Progressives and Republicans could work out a deal, alternating the presidency and VP slots and not competing in each other's congressional seats (Australian Coalition style, but with equal-ish parties).
Perhaps this leaves the GOP able to really get into the Southern conservative support, and the Progressives to pick up House seats on the basis of the black vote (once people get over the name ID problem).
I can see their potential voter coalitions fairly easily, I'm just not sure how to prevent the framework of American politics forcing everything back down to two parties—especially with the 270 electoral vote requirement.
Furthermore, once the Great Depression rolls around the Democrats freakin' dominate the way nobody ever has—what does that do to the Progressive Party? (Of course, butterflies.)