I just escaped San Diego a few months ago. Many people don't realize east county gets snow, although it's usually melted by afternoon.
Fortunately for you, or maybe I should say unfortunately, I grew up in Salt Lake City, UT. You got the wrong wheel and tire setup for snow and tire chains. While everybody else is going to warn you about destroying the paint on the edge of your fenders (and they're right), I'm going to warn you about scratching your wheels. You need less wheel and more tire. Did you even get a 4wd truck? Something tells me that's probably a factor in this too. If you know you're going to venture where there is snow on a regular basis, getting 4wd, and the proper tires for 4wd, seems like common sense to me. Big Bear is over 2 hours north of San Diego. After driving that far I wouldn't want to deal with the hassle of installing chains. Pushing a button is just as effective.