To get a BA in engineering, I believe that the highest level of math one would need is Linear Algebra; so some of the courses required would be Algebra, Trigonometry, Pre-calculus, Calculus, Differential Equations, Linear Algebra, and Statistics at the Engineering level. You will not have to do any rigorous proofs, but there are a few in Linear Algebra, and identities and a few small proofs in the other courses. I can be wrong on this, since I am not an engineer, but a recovering math major (I was a math major and got a B.A., but had a nervous and mental breakdown from too much stress...long story).
Out of Biomedicine, Architecture, and Engineering, the path with the least amount of math I think is Archetecture, as one only needs Calculus 1, although some biology majors (depending on the school and perhaps the field of study), only require Calculus 1; some require Calculus 2.
Please understand Salome, that I can certainly be wrong on this, but as of 15 years ago, this was laid out for me by a counselor in college.
No matter what path you take, I hope the most success for you and that you end up happy doing what you love.