I believe the Sugar Bowl is obligated to take the winner of the championship game unless they are in the playoffs, and then it is required to take the loser of the championship game (unless they also make the playoffs?).
This site:
https://www.dish.com/dig/sports/explaining-the-sugar-bowl-selection-process/
states it is the 2nd place team, BUT that was before the Big 12 added a conference championship game. I could not find any update since, so it seems likely to presume it would still be 2nd place, not the Sugar Bowl's choice.
Interestingly, the article notes the Sugar Bowl contract is different with the SEC, where the SEC will send the next highest ranked team in the CFP rankings (which still is not the Sugar Bowl's choice).
I think the key difference is that the SEC has two divisions. So, let's say, the championship game was:
West Champ: 12-0
East Champ: 9-3
... and the runner up in the east was 11-1. The conference wanted the 11-1 team to go over the 9-4 team (if the East were to lose in this example).
The B12 can have similar examples based on how teams perform out of conference. But much less likely given the fact that every team plays against each other. But it can happen.. For example, this year, if the Sugar Bowl must take the loser of CCG assuming winner is in playoffs (long shot at this point that OU would make it) ISU could go the Sugar Bowl as an 8-4 runner up, while W could be left out at 9-2.