Yes, but why go to the trouble of forming the third party in the first place?
Note that the Bull Moose party was a “Connecticut for Lieberman” scenario. Under our system the parties don’t choose the candidates, the candidates choose parties. Then the primary voters chose the party representative. So the only time your white knight needs to form a third party is if they have lost the primary. But the thing is, if you’ve already lost the primary you’re in big trouble, and the most likely result of your third-party is to guarantee the election of the other major party candidate.
The only reason this didn’t happen to Lieberman is that in his case the Republican candidate was just a placeholder. That’s never going to happen in the presidential election, even if the incumbent is overwhelmingly popular the opposition party is still going to do its best, even if it’s pretty clear they’re going to lose hard (cf Reagan vs Mondale).
A similar situation occurred in Alaska, where Wally Hickel won the governorship on the Alaska Independence Party (note the -ence, not -ent, this is the secessionist party that Sarah Palin’s husband was involved with) ticket, against weak Republican and Democratic candidates. But he quickly rejoined the Republicans, and everyone knew he was just using the AIP for ballot access.
The point is that if you’re a center-right White Knight, you can just run as a Republican, and if there’s really a groundswell of pent-up demand for a moderate against the populist moonbats, you’ll win. And then you’re the leader of the party, and you dictate the party platform. Or rather, the party platform is irrelevant, what matters is your personal platform. And once you’re the nominee, the congressional candidates fall in line, or you kick their asses.
And this is how party realignments happen.