Biochemistry would be a good alternative major since it is not capped at UCSD.
In general, if the major is in the College of Letters and Sciences, then you are NOT admitted by major.
Below is how each UC campus handles alternate majors when applying.
UCB:
Division (L&S, CNR, CoC, CED, CoE) matters for admission selectivity.
Within CoE (but not the other divisions), major matters for admission selectivity. Changing majors within the CoE after enrolling is not guaranteed, unless one is CoE undeclared.
Alternate majors are not considered.
UCD:
Applicants are encouraged to list an alternate major, but not in the same area as the primary major (e.g., Computer Science Engineering with alternate Computer Science in the College of L&S). Occasionally, we admit to the alternate major.
UCI:
UCI admits into the University first and then into the major. In the case that UCI is unable to accommodate all qualified applicants in their first-choice major, those students who indicate a valid alternate major may be offered admission in that major or Undeclared.
UCLA:
UCLA only guarantees review of an applicant’s first-choice major.
We typically do not admit to the alternate major.
UCSD:
The campus does not admit students on the basis of academic major or choice of UC San Diego undergraduate college. Alternate majors are considered and capped majors are highly competitive. Also note: Capped majors require additional pre-req courses and specific GPA to be able to qualify if changing majors. Also if applying to a capped major, select an non-capped major as an alternate.
UCSB:
College of Engineering: Students are selected by major for all engineering and computer science majors. Only applicants with a solid background in advanced high school mathematics will be considered for admission to engineering. This includes high grades in all math courses through grade 11 and enrollment in pre-calculus or higher in grade 12. A student not selected for their first choice major will be reviewed for admission to an alternate major outside of the College of Engineering if one was selected.
UCSC:
Important Note for Prospective Engineering Students: Choice of major does not influence the selection of first-year students, except for those applicants interested in a major offered by the Jack Baskin School of Engineering (BSOE). Freshmen who are interested in a BSOE program should be sure to indicate a BSOE proposed major. Students who do not indicate a BSOE program or who apply as undeclared might not be able to pursue a BSOE program.
UCR:
Admission by major but alternate/2nd choice major will be considered if applicant does not need their first choice admission standards.