In this case, they absolutely will. The committee will not, under any circumstance, put a 2 loss team in the CFP over a name, 1 loss program who’s only blemish will have been to the 1 seed (Which LSU will be under your scenario).Alabama doesn't have a chance for a signature win. Beating Auburn won't mean much, and the committee won't punish Georgia for playing an extra game and (presumably) losing to #1 LSU.
There’s no chaos needed here, this is clear cut. The only chaos comes in the event of another Alabama loss.