No, it pretty much works this way: If you win the National Championship game, you are the national champion. It is winner take all, just like in every other sport.

If a NFL team gets beat by a team, and them beats them in a rematch in the Super Bowl, should they split?

If a baseball team wins the overall season series, and then lose to that team in the World Series, should they claim 1/2 of the World Series?

Does any other sport prevent a team from playing a national championship because they didn't win their conference?