It would have been easier to say right away that single-player games don't belong in an organized tournament.
You, the person who is trying to organize a team tournament, are not ready to play with another opponent.
What kind of promotion of team games are we talking about then?
And the fact that someone is not ready to play with another person is a completely normal situation. And the fact that some teams won't switch is also normal.
In short, in my opinion, when there's 1 team game + 2 single-player games, it's fine, but when there's 1 team game + 4 single-player games, it's not fine because the value of the team game is greatly diminished, and that's where, in my opinion, all the fun lies. So, the format where 2 people play a team game and 2 play single-player games is the best.