I think you should contact UHP and ask if being in the program would help with enrolling in restricted courses such as CS. Tell them that you are weighing UC Davis against Berkeley.
UHP usually has an admitted student event. Are you able to attend that?
Edit: Davis offers CS in both Engineering and L&S.
The L&S CS website has information about how to declare CS as a 2nd major.
UC Davis also offers a minor in computer science. The UC Davis CS minor page says,
Several majors at UC Davis take classes within the Department of Computer Science, such as cognitive science, applied mathematics, applied physics, biotechnology and other majors in the College of Engineering. These are all viable options for students who want a career in the field of computer science.