I have two thoughts on this, and I certainly don't mean any offense by either (note - My top 5 schools rejected me too, and I'm not thinking about reapplying)
Thought 1) Yes, Stanford is better than UCLA, but you're postponing your MBA by a year. Since you have a JD already, you're probably older than the average applicant (or at the very least equal in age), and if you do really need an MBA for what you want to do, I think you shouldn't hold off by a year. If you truly need the MBA for whatever your plans are, what do you plan to do this year, other than stress out about Stanford's application? It sounds like you're willing to burn a year of your life for a shot at a school you're a heavy underdog to get into.
Thought 2) Reapplicant admission percentages are low. On numbers alone, you're a decided underdog to get into either of the schools that rejected you. Sure your story might be better, but it's a fairly big risk. I don't think UCLA and USC are total shoe-ins after you turn them down (to not go to another school). You turn a school down to go to school somewhere else; it happens all the time. You turn a school down to reapply next year to give yourself 1 more shot at your dream school; it's rare and if anything shows you don't want to be at UCLA, USC, or UCSD (who probably will take you without issue). To me, it's a bit weird.
Anyway, Stanford was my dream school too, so I know where you're coming from. And after Stanford, I wanted to go to Chicago. For me, holding off by a year doesn't make any sense. I don't think I'd get in, and I'd piss off the schools that took me, so that I can use them as safety's next year... I want my MBA, I think I will help me for my career, and I don't think there's any value in waiting.
I think you need to think long and hard to prove that you'll get in next year, and that a UCLA MBA has significantly less value to your career path than a Stanford MBA. I was in LA 3 weeks ago - UCLA is a beautiful place. I wouldn't be so quick to turn my back on them.