Found a little information here, which you probably have already seen, which says you can submit a change of major request one day after fall enrollment, or later, using the Major/Minor tool on TritonLink. They say it is first come first served until the numerical limit is reached. It doesn’t really have the information you need, which is how likely is it that the request will be granted with things the way they are.
http://biology.ucsd.edu/education/undergrad/admission/capped-major.html