Stuck between Undergraduate Engineering Choice - Need words of Wisdom

the usual way is to get a BS/BA from the “pretty good university near where you are” and follow it up with an MS/MA from “some US university good enough to get recruited by a visa sponsoring company.”

Variations exist, but from what I’ve seen, it’s pretty rare to get a visa/green card sponsorship with a US undergrad in engineering. I imagine it’s even less common for business and humanities majors.