I think it was @clarkwgriswold that once called the winner of the NIT, the best of the losers. That stuck with me, because it’s so true. I don’t think the NIT could ever draw significant attention during the NCAAT. Personal and institutional pride gets in the way of everyone trying their hardest.
But if I were to try, I would avoid bragging rights or pride altogether. Hold the event at a unique or special location and give financial reward for every game you advance. The money could go to the players or in the team’s collective. I remember Michigan State once played North Carolina on an aircraft carrier. That was must see TV. And I always liked the Maui Invitational. Something like that with only 16 teams could draw some interest for fans, players and the general public.