Private schools in EG have been a topic for years. My recollection is that there are none that are deemed decidedly better than public schools. You will not know nearly as much about them from data, as their reporting and testing requirements are minimal compared to the public schools.
Some of the private schools claim a higher college admission rate, which is not surprising, given that attending those schools costs money, is elective, and as such tends to cater to people where the parents are involved. That does not mean the education is better than a comparable public school. Given that teachers in secular and religious private schools make less than 60% in comp packages of what public school teachers make, they often do not get the pick of the litter and in fact do sometimes get the rejects. There are a handful of private school teachers that elect to take less as they feel that strongly about the school, but that’s the exception over the rule. Just know the land mines going in, and don’t assume because it’s private, it’s better. Usually it is not, but there are some diamonds out there that fit the needs of the student well.
Have you looked at the Montessori school? We had our daughter in one until the 3rd grade, and really liked the approach. It is the opposite of Waldorf–it’s structured and task oriented. The only reason we put her in public school was the classes got too small. I think her 3rd grade class had five in it. Not all Montessori schools are good or well run. You might check the one in Elk Grove, talk to some parents, I’ve not heard anything about it.