Once elected, a new President would lose his/her original party affiliation as s/he is now the leader of all citizens.
If s/he stands for re-election, it would be as an Independent candidate competing against a Democrat and a Republican. There would be no need for "running" a campaign prior to the election. The President's public record of the previous four years would be the guide for the voters.
Also, there would be no need to change the Constitution or set up a third party. A simple change of "job description" for future presidents would suffice: from a party-driven mandate to a country-leading one.