I always thought most condos in Downtown LA were not VA Approved. I was wrong. I just confirmed that Luma South (1100 S Hope St) is, and I think if it is some others will also be.
If you are not familiar with VA loans here is a bit of info: https://www.va.gov/housing-assistance/home-loans/eligibility/
One of the biggest attractions is the Low Down Payments.
Luma has units starting from $550,000.
Let us know if you would like to see some.