Auburn should come in at under $40 k with the presidential scholarship. Out of state tuition is $35k, housing and food if living on campus are about $17k. Last year with a 35 -36 ACT the highest scholarship was $17k. I’m no mathematician, but that would come out to $35k plus whatever is needed for travel and other incidentals… And because I know you are going to say it - yes, tuition will likely increase next year. Yes, the amount of the scholarships for next year have not been confirmed yet either. That’s why I said it “should” be within budget. I would also note that most students live off campus after freshman year for much less than on campus housing costs. My son paid $600 a month including utilities for a place within walking distance of campus. He also had smaller departmental scholarships that stacked with the presidential. Definitely not as cheap as Bama and maybe not as prestigious as OP would like, but as I said Auburn does have a good formula SAE program and is a little higher ranked and probably less expensive than Arizona so it might be of interest because of that.
Edited to add: Purdue should absolutely be considered but will most likely be over $40k. Georgia Tech might be another one to look at as a moon shot. It would only come within budget with the Stamps scholarship. My S22 has several friends involved with the formula SAE team there.