I graduated with a BS in molecular and cell biology and a GPA similar to yours, worked a few years in a government research lab, took two math classes as a non-degree student (differential equations and multivariable calculus), applied to a number of thesis-based MS programs in mechanical engineering, and was admitted to several top schools. Though I’d self-studied most of the fundamentals of an undergrad engineering curriculum before applying and, as a biology major, had already taken two semesters each of calculus and physics (as well as statistics), I hadn’t actually taken any engineering courses. In hindsight, I’m surprised I was admitted anywhere at all. I have some guesses as to why I was admitted, but my case is an exception nonetheless. After matriculating, I struggled to keep up at first but ultimately succeeded, so I guess the members of the admissions committee made the right call.
I’ve met a lot of other people who also transitioned to engineering (including mechanical) from the life sciences, so it’s certainly doable, but the most reliable way to do so is what the others have mentioned: take the prerequisites. Multivariable calc, linear algebra, statics/dynamics, thermodynamics, fluids, controls, etc. You don’t necessarily need to take all of them, but enough to show that you can handle what you’re getting into.
Career changers often take these prereqs as non-degree students, but I think it would be easier to do it while you’re still an undergrad (which you said you’re already doing/planning to do, so that’s good), even if that means taking longer to graduate. If, for whatever reason, you can’t take engineering classes, then take as many math courses and quantitative science courses as possible—physics and math majors have relatively little trouble getting into engineering graduate school. I imagine a biology major with a strong math background would have a similar experience.