“I might have the option to ask admissions to consider me for Biochemistry instead of Bioengineering,”
I think that I would apply to the major that you actually want.
Your major impacts what you do in your career. Which university you attend just impacts where you are for four years. Also, the “lesser UCs” are still very good universities, as are Purdue and Penn State.
Assuming that you applied to the major that you actually want to study, I do not think that you should worry about it.