Daniel, if someone doesn't work in Television, they might not understand your position. I'm an Editor in L.A. so I totally get it. My advice is to either purchase out of state as other have suggested, or keep stacking your cash and growing your career. If you're not already a member of your union guild, I would join ASAP and focus on that as much as possible because your income can quickly double and then you'd be in a much better position to purchase in Los Angeles (or anywhere else for that matter).
As for getting loans here, local banks and credit unions understand multiple W2s (especially when the come from Cast & Crew or EP) and inconsistent work in your field because Television is the largest industry here. Talk to someone at your bank and begin to form a relationship there so that you can find out what they'll be looking for from you. Also talk to your union about what banks might be best to work with and DEFINITELY go to First Entertainment Credit Union if you haven't already. Maybe the best way to do it if you have your heart set on starting in L.A. is to house hack. Check out Naca.com to find a good deal on a loan for your own home and see if you can get a duplex/triplex/quadraplex. Good Luck! Take what I say with a grain of salt as I'm a slow learner and just finally decided to purchase out of state after nearly 6 years of trying in Los Angeles. Looking to close on my first property in Oklahoma in July.