Voting for a third party will push red and blue to change more than asking them kindly.
Actually, it’s probably the exact opposite. They want fewer people actually involved in the election, because then they need to convince fewer people to vote for them in order to win. If you effectively remove yourself from the voting pool, by voting for someone who can’t win, you are basically making it so they don’t have to court you, not that they will.
Even if you managed to get a third party elected, it would just revert back to two parties very rapidly. This is the nature of a FPTP voting system. This is not something you can do top down by flipping a switch. It’s gotta go the other way. Work on your local elections.