How big of a deal is it if I abbreviate a school name?
I can't imagine that anyone would object, as long as the abbreviation is widely used by people who work at the university.  (UCLA is fine, but I wouldn't call it UC Los Angeles, since I believe that's a much less common abbreviation.)

In fact, repeatedly writing out "University of California, Los Angeles" is more likely to stand out as strange than using the abbreviation UCLA is.