There’s only 1 way to do this to make it work. P5 get 12 each, whatever the next level down gets 8 each, small conferences get something like 4 each. Make it regional.
You get 128 teams into the tournament. 3 games series every weekend for a month. Higher ranked team gets the series at home until you get down to the last 8 in Omaha.
Yeah this schedule takes it into July, but only for 8 teams. The conference tournaments help determine the seeding. #1 seed gets #32 in their region, should be a cakewalk. #16 and #17 teams in that region are about the 64th to 71st best teams nationally, kind like Miami possibly. Everybody gets a chance, lots of extra home games for the top teams. After 2 weeks you have best 32 teams nationally minus the handful of cinderellas that won their way in.