With all due respect, Auburn's elite women athletes are proud Auburn Tigers, full stop. Just like the men. Although I know it's not your intention, the 'Lady' designation feels condescending to me (and, I suspect, to Auburn's women athletes).
The only university I'm aware of that officially uses the 'Lady' designation is Tennessee. In 2014, they phased it out for all sports except women's basketball:
https://swimswam.com/university-tennessee-phases-lady-vols-name-womens-sports/
But in 2017 they brought it back:
https://www.espn.com/womens-college-basketball/story/_/id/20704564/tennessee-brings-back-lady-volunteers-name
No 'Lady Crimson Tide', 'Lady Bulldogs', 'Lady Gators', 'Lady Gamecocks', etc. - and, please, no 'Lady Tigers'! Thanks and War Eagle!