I've been thinking about this one ever since the tournament became a possibility. Group stage games are regular season because they count in the regular season standings. If we make the "postseason", I think I'll make those as Playoff games, sort of like Merconore and CCL are mushed into International (and Leagues Cup would have been). I don't want to create a new game type for a one-off tournament.
I suppose you could just as easily classify them as Open Cup, or simply just "Cup" games, as there's almost certainly going to be no Open Cup this year, whereas there's a better probability of there being some kind of MLS Cup and MLS Cup Playoffs in November, December… the league hasn't ruled out having it in January of next year!