I had a couple that did exactly that, chose USC over UCB and UCLA. We like the perks of a private education and in particular the ability to shape your education exactly the way you want to during your time there. Changing majors is simple or double majoring even with a minor is very do-able in four years at USC (and particularly easy for students with high gpa’s that get perks of taking additional credits without additional cost). Relationships/connections made at USC and the alumni network are as strong and valuable as the education. Engineering project and research opportunities during the four years were crazy good. So those were our big draws for USC over UCB. But truth is, mine never connected with UCB on the many visits we did there, so it was never that tempting.
You mention UCB is your dream school and that carries weight. Years from now, if you want to say you graduated from UCB then go there. You can excel at either school and both are fabulous in their own way - so go where you feel you really want to be, simple as that. Good luck!