Can’t you setup a structure where people from both sides would together commit to voting third-party?

Can’t really do that with a First-Past-The-Post system because someone needs to get past the post. If there are four major parties (left and right both have significant numbers voting 3rd party), there would ultimately have to be a coalition or two that just ends up being the Republicans and Democrats all over again.

The system itself has to be changed first, and the two parties who benefit from there only being two parties aren’t going to change it to allow for that.