I’m right there with you. I applied to a school yesterday morning and last night at 10 pm I get a decision; denied. They went through my application in less than 24 hours and handed me a rejection. I was speechless. What the f was that?
I wouldn't feel disappointed about that one rejection you got. Some schools reject people who are over-qualified because they think those applicants won't even attend if they get accepted.
I think you have a very good chance, cause if you are below their 50% it doesn't matter how much you below. Also, ucla is trying very hard to improve their ranking i guess, they will need some high lsat taker.
I don't think you can apply without at least one (or two) LORs. I think it would also mean an automatic rejection, but I could be wrong. Is there someone you are close to who you trust and is ranked higher than you who could write you an LOR?