Despite what many people will say, a degree in physics and a degree in nearly any engineering discipline will be fairly different, especially electrical engineering. I am a physics major at UCSD, but i have taken the lower division physics series that engineers take (its pretty interchangeable here), and i would not worry too much about class sizes and if you still were worried the physics major series (the 4 series) usually has about 20 students in it. I have never felt that the size of a class has hindered my ability to learn. You can still talk to the professor after class and at office hours and the large classes usually have multiple TA's. Office hours almost always have 3 or fewer students and many times no students at all, so i would not worry about having too many people your classes, plus most upper division classes are small(ish) so the large classes are for the beginning classes only (for the most part).
If your plan is to get a job in industry, as opposed to academia or basic science research, engineering is the way to go. I could tell you how a bunch of electronics work (the basic electromagnetic principles and theory behind the device) but i couldn't tell you how to build the devices, so if you want to actually build things engineering should be your focus. I have friends who are EE majors and the kind of stuff they have to do is very different from what we do (once you get to upper division that is), not to mention all of the classes we have to take that do not apply to EE. Don't get me wrong, i love physics, but if you want to be an engineer it is not the major you should use as a substitute.
There is also something to be said for actually having an engineering school and all of the resources and opportunities that having an engineering school brings. In engineering, the way to distinguish oneself is to get engineering experience, usually in the form of internships, and having a school that will help you get these (from engineering firms or professors) is a big boost in getting a job down the line. Between the military (BAE systems, lockheed martin etc), biotech, and telecom (qualcomm) San Diego is an engineering rich city so having a degree from a well respected and near-by school will really help. Plus then you set yourself up to live in San Diego.
UCSD's overall undergrad prestige may not be as great as UChicago's but UCSD certainly has a good engineering reputation. Even the east-coast-biased, prestige-obsessed usnews rankings recognize UCSD's engineering school as number 12
Rankings - Best Engineering Schools - Graduate Schools - Education - US News and World Report. The graduate rankings are indicative of where the undergrad rankings will be going in the future, plus UCSD has several BS/MS programs where you can spend one extra year and get your masters (which usually take 2 years) so there you can also save some time and money.
I think the money saved will really help in the long run, and you can turn that money around and invest it in grad school (because you are going to have to pay for engineering grad school somehow). UChicago is a great school, especally for physics (i wish i had applied), but between the money and the academic fit (there are other fit criteria i am ignoring here becausei do know more about you) UCSD makes sense. If you wanted to study physics and the price difference was a bit smaller i would be singing a different tune, but despite UChicago being a really good school it sounds like UCSD has what you need and UChicago does not. Good Luck with your decision.