You can't win without paying for it

There's a reason Duke, Louisville, Michigan State, etc are good in basketball. They pay their head basketball coaches multiple millions of dollars each year. They have coaches worthy of that salary, however. The results are satisfying to the players, students, fans, administration and boosters. It's not cheap to be in the tournament every year.
So, until LSU is willing to pay a head basketball coach a salary comparable to Les Miles, then expect to remain mediocre every year competing for NIT bids. Maybe it's just not important enough, because TAF damn sure has the money. The coaches are out there, and they would come to Baton Rouge if LSU really felt it was necessary to be in the NCAA tournament every year.

FYI, Wichita State pays its head basketball coach, Gregg Marshall, $3.3 Million per year. I wouldn't mind having him here.