Shopping should be removed from your equation since there's not much you can get in LA that you can't get in the UK, its just that "stuff" is usually cheaper in the USA, so its more of a decision of whether or not to buy "stuff" and that has nothing to do with your trip. Eating out should be much cheaper than Europe, $50/day should be plenty, unless you gravitate to those kinds of places where they give you a tiny dollop of something in the middle of a gigantic plate for $100. Try some mexican restaurants - most are really reasonable and very good.
The thing you are really misguided on is transportation, for most of the USA and especially LA, a car is the ONLY option. Yes, there is the illusion of public transportation, but if you are going to rely on it then you will be spending the majority of your time waiting for and sitting on buses with little time left over. LA with all its surrounding "cities" is HUGE, you can spen half a day just getting from one end to the other in a car! Taxis are expensive $3 to start and then $2.70/mile - that adds up quick, and you'll need a cell phone, but you can buy a prepaid one for about $30.