I'm against expanding the NCAA Tournament, but what about...
The winner of the NIT gets an automatic bid to the NCAA tournament next season. At worst, the NIT winner replaces the 11-seed in the play-in game, but then they can be "upseeded" if their play warrants it.
Mentioned in another thread, but European soccer does something similar with the Europa League and Champion's League. Teams used to phone it in for the Europa but began taking it more seriously, especially as they got closer to the title, once that change was implemented. Sure it would make teams think twice before declining. The new Fox tournament is awful, however.
It'll dilute attention, while still being in competition with the NIT, so both are weakened. And it's between the Big East and other big conferences, ensuring that weak teams from established conferences get more screen time instead of interesting and dynamic teams that are, admittedly, less familiar. It's a mirror of what is happening with conference realignment, in the worst possible way. The NIT might not be long for this world with the Fox tournament. And I think the Fox tournament, lacking the charm of drawing from the entire pool of teams, won't last long either.