When you say "Center for Autism" do you mean David Black's shop, CAAT? If so, Dr. Black has worked with plenty of families at Maddux and everywhere else, and could give you good advice about where would be a good fit. "Mild" and "moderate" labels are tough, because all of our kids have strengths and weaknesses that are hugely diverse and vary not just by situation but also from one day to the next. And our society is so judgmental about social differences that no form of autism is a mild problem, really, in the sense that a physical limitation or stutter might be. Nobody uproots their family and spends $30k+ for a special needs preschool for a problem that is actually mild.
It's possible that buried in one of your reports is a reference to a "level" between 1 and 3, which is how the DSM-V classifies the severity of autism. Those descriptions are here:
https://www.autismspeaks.org/what-autism/diagnosis/dsm-5-diagnostic-criteria
There's a lot of diversity at Maddux, including kids whose issues are language or anxiety based, kids who have some challenges but don't really fit a diagnosis, and even typically developing siblings. I think you would describe the kids who have ASD as L1 generally, although I think that almost any usually-L1 kid can look like the L2 description on a bad day or in a bad moment. The best way to get a sense for fit is to go visit. They have tours and open houses this time of year, and I suspect that just walking around you'll get a sense of whether you can see your child there.
Best of luck.