Would you recommend expanding my search area, wait it out, or look into duplexes in my current markets? 

I was really interested in a 1/1 on top of a hill (had a bit of a view) but was several miles north of Hochatown, so was talked out of it by my realtor.

Get out of your comfort zone.

I assume you can be taken out by an existing investor or using a broker to find a buyer for your stake if the asset is a good cash-flowing one?

Short-term rentals would be a solution, but due to strict short-term rental laws in rent-controlled areas of Los Angeles, options like Airbnb are typically not feasible.At present, I have a $5 million property in Encino with two units that we cannot rent out short-term due to rent control regulations, and the numbers don’t work for a long-term rental either.

You can't even try out the calculators to see if you like them. 

Our mortgage agent suggested to buy as an investment and then refinance, but is not a good idea since we cannot take cash out (we don't have enough for the 25%).We thought of sending an offer contingent on one of the units being vacant, but the seller doesn't want to put that into written. 

this would be exceedingly rare to find any PML who puts their money out for 30 years.there are note investors of course that buy 30 year paper as in defaulted notes but they are paying a very big discount to get them.

My wife and I are currently saving for our first property to rent out.

What are the my options to limit bringing that money out of pocket for the down payment but also minimize the amount of equity given away on the project?