The winning record requirement for the NIT was dropped in 2017.
Here's one way for Mizzou to finish 10th and avoid the first day of the SEC tourney:
Mizzou beats Alabama
Texas A&M beats Arkansas
Mississippi State beats Ole Miss
Florida beats Kentucky OR Auburn beats Tennessee
That ensures that Missouri and Arkansas have the same league record. They split their season series, so the next tiebreaker is record against th second-place team. Mizzou beat Auburn and Florida, so either of those teams finishing second would give MU the nod. If both Florida and Auburn lose Saturday and LSU beats Georgia, then LSU is second and Arkansas gets the 10th seed based on its win over LSU.
Basically, the Tigers need to be in a tie with Arkansas and need Auburn or Florida to finish second.
A three-way tie with Mizzou, Arkansas and Ole Miss for 10th place will always favor Arkansas because of its 2-1 record vs. the Tigers and Rebels.