If I were in your shoes, I'd change my shoes and then focus on setting up and optimizing a Google My Business profile. If you've already got that, good, on to the main question...
Does google recognize city abbreviations in page titles?
It can, yes -- and in your situation there's no reason to doubt it will, LA is an incredibly common abbreviation for Los Angeles.
I'm wondering if I can't get away with LA County in my homepage title.
You say you're a small local business, do you serve the entire LA County??
- Yes; go for it.
- No; don't go for it.
I assume it's important for SEO to have that in the title.
Only somewhat, though it is a quick way to inform/reassure users of your business' location.
Will LA County have the same impact as Los Angeles County?
It's likely, but there's no way to know without performing some A/B tests.
How about area codes, like 818?
I doubt that'll perform better, unless that's what your audience searches for.
Overall, I suggest worrying less about Googlebot's capability for interpreting title tags and more about creating a unique brand persona and/or value proposition. Optimizing title tags alone can only get you so far.