Many small regional LACs in the Midwest give a limited number of full tuition scholarships. These may not be “national names”, but they provide a high quality education with a lot of personalized attention and mentoring. Here are just a few I know of:
Carthage College: Quite a few full tuition scholarships. A very strong student from our local high school went there and is thriving. Within 6 hours of STL.
Gustavus Adolphus: about the same distance from you as St. Olaf, and a lot of the same vibe, while probably more likely to get you to budget.
Ripon: I know a National Merit commended student who nabbed one of their competitive full-tuition scholarships. They are starting an engineering major in fall of '24. It won’t be ABET accredited yet, but is designed to get that accreditation, and then graduates will receive ABET accreditation retroactively. Engineering can be a terrific pre-med major (a number of my medical school classmates had undergrad engineering degrees–mechanical, civil etc.)
UMN-Morris. Minnesota’s public LAC. About 10 hours away, but a very good biology department and great mentoring. Baseline OOS price is <30K, and your student would likely get merit scholarships on top of this. In addition, cost of living in the area is really low. I know a number of accomplished grads including a biology professor, a diplomat, and a number of doctors.
Out of the box idea: UMN-Rochester. Not a traditional college experience, but an incredible opportunity for the right kid. Small college that offers only pre-health professions majors. Right there by the Mayo Clinic and Mayo Clinic Medical School so a ton of research opportunities and mentoring. OOS tuition is same as in-state, and there are multiple scholarship opportunities on top of this.
ETA: I would not worry if a school does not have a separate Biochemistry major in addition to Biology. The college I went to was a well respected LAC and had no Biochem major. Despite that, I graduated with a very strong foundation in biochem.