I've had the chance to speak with many, many prominent people in all aspects of entertainment, from agents to studio execs to in-house counsel to heads of business affairs, development etc. etc. (all of whom were USC Law Alums). The general thread seems to be: know someone. Some of these people had family/personal connections, and some of them just made their own connections in LA. They all seem to say to do as well as you can in law school, get a job in a relatable field in a reputable firm, and then lateral your way into something entertainment-related through your connections. A few people do push through by taking internships while they're in law school and getting right in (depending on what you want to do), and I do have some friends who are doing entertainment gigs at NBC, Warner Bros, and firms that handle entertainment, but honestly that all just speaks to "knowing someone".
The reason I push USC (besides the obvious fact that I go here) for entertainment is that you're making connections where you don't have any. We're fortunate enough to have incredibly powerful alumni who are big players in the industry (plus we have a few current students who have very close personal connections with huge players in the industry). This is how you start building your network, because of the opportunities that going here presents. It's less to do with the classes/certificates/etc., and more that powerful people are willing to meet with you and talk to you while you are a law student just because you go to USC, so that when you are a lawyer, you will know powerful people that can help you into the entertainment industry.
To the OP: I know a guy who has very similar aspirations to yours. PM me if you want me to put you in touch with him, he's all about the music industry and has landed a job with a small firm in Beverly Hills that does music law specifically.
(Applications Advice, Letters of Recommendation . . . )