I don't know it is at UCSB, but at my UC they were REALLY strict about students not being able to stay an extra year. I think it's because of California's budget disaster, but they essentially force everyone to graduate in 4 years, and if you can't (no matter what your reason), then you're forced to take summer school the next year to finish. Even if you wanted to pick up a minor or double major, you could only do so if you could prove that you would finish everything in 4 years max. I tried to pick up a Spanish minor so that I could stay an extra year and take pre-med pre-reqs, and was flat out rejected because of this 4-year-max rule. The school made it almost impossible to add a major or minor if you didn't do so before a certain time.
I also know for transfer students at my UC, they were required to finish their degree within 3 years of transferring. You might want to look into this further for your school.
If worse comes to worse, you can always apply for a post-bac. An academic-enhancer post-bac (or those geared towards URM/disadvantaged, if you are such) will allow you take undergraduate-level courses, which I think would help you more than a master's would at this point. A master's program won't let you to take/re-take undergrad courses, even though they do have other benefits.