Request to move-up interview
Business schools operate differently from moreal traditional departments. That said, tenure-track offers still need to generally be signed off on by a dean. Even if you interviewed for the tenure-track position tomorrow, and you are the last planned interview, the offer could still take a couple of weeks.

My opinion is that a tenure track position is enough of an upgrade over a non-tenure track position, that you are justified leaving at any point. Non-tenure track positions are abusive to young academics and if a department won't make a multi-year guarantee to you, I believe you can leave whenever you get a markedly better offer. In that light, I suggest you let the tenure-track process play out withough you trying to speed it up. I suggest you slow roll the non-tenure track offer and if they force you to sign before you hear about the TT, you do, then if you get a TT offer, you back out.