On one hand, he is screwing over his current employer right when he has them in a position to make a run in the playoffs, a position they have never been in before. There is no way this whole situation can not screw up the team at this time.
On the other hand "Ole Miss knew exactly the kind of person Kiffin was when they hired him, so they need to take the bad with the good. Ditto for LSU.
Is there a good reason LSU would make this move now, rather than waiting?
I suspect they are feeling urgency for some reason, to such an extent they would risk the post season now in exchange for the expectation of better performance next year.
I think you are getting your schools mixed-up. LSU is not going to the playoff this year. They also fired their coach mid-season and are looking to replace him. The early signing period for recruits starts in December, so schools want to have their coaches in place by then.
Ironically, Brian Kelly, who LSU just fired, came to LSU from Notre Dame in the exact same manner Lane Kiffin is now. LSU pays it coaches well and gives them a good shot to win a national championship, but in recent history they have eventually chewed-up and spit-out their coaches in the end. It will no doubt not end well between Kiffin and LSU either.