Out of curiosity, which mustachian candidate cities have similar weather to LA? I'd love me some of that weather without the cost.
So... something of a heads-up: there's a reason housing costs more in that sort of weather. Demand and all that.
I'm probably hurting the "I'd like my eventual kids to be able to play hockey" line of thinking, but Phoenix (AZ) is currently my top choice. But - and this is a serious warning - you have to like the heat come summer. Not only do I, I've regularly gone to there and Palm Springs (CA) in summer for vacation. Also, houses seem far more reasonably priced than in Los Angeles.
I also like Irvine (CA). It's just down the coast from L.A., very much able to be biked around, very safe, maybe a little pricey in the housing department (I haven't checked).
Would either of these cities top MMM's list of mustachian cities? Almost certainly not. But both have decent public transport and can be reasonably biked around (be sure to bring plenty of water in the summer, especially in Phoenix). Both have inexpensive entertainment options. Both are places where getting solar panels will probably make sense and pay for themselves quickly.
To me, part of the motivation of being frugal is having the money to spend on the things you want to spend it on - and for me, living in these conditions is worth that I'll probably have to either retire a few years later or work part-time in the first few years of my retirement. If I can find a part-time job I can enjoy in one of these cities, all the better towards that goal. But my non-housing expenses - even in Los Angeles - are so low that I'm not really worried.
Long story short: it's not as low-cost as many other areas in the country, but it can be done. It's all a matter of how much you're willing to spend (or, how much of your life you're willing to work to be able to spend the rest of it in these places). Only you can decide if the trade-off is worth it to you.
There are also parts of Palm Springs - and Riverside County in general - that is pretty inexpensive, but investigate the neighborhood before being too excited about it. Some are good, some ... aren't.