How about we start with a just few 3rd party congressmen? That can't be too hard, right? 3 or 4 congressmen not beholden to either party. I'm sure there's at least a couple districts out there that the majority could be swayed if enough focus was put on them.

Then we elect a few more.

Then maybe a senator.

Now suddenly this 3rd party starts getting some attention. It's still by no means a majority, but that's enough congressional votes to give them a seat at the table. When votes are so split and close these days, even a small number of votes can throw things out of order. Demands can start being made.