OP - We recently went through the process of deciding between an excellent McLean public school and a private. I have a few recommendations:
1.) Give some thought about DC's personality, your educational goals, your/DH's educational experiences, your family logistics, the dynamic in your neighborhood (how many kids go to SH vs privates?), your DC's learning style/needs, and of course, your finances. All these things help factor into "fit" and then help drive the questions you want to investigate. How do you feel about homework in the early grades? What are your child's strengths and weaknesses? How is your child socially? What worries you about your child? How willing/able are you to supplement your child's learning if he/she is struggling with something?
2.) Tour SH and the privates, but talk to lots of parents. Ask specific questions about what the child is learning, how the "specials" work - library, PE, etc, what the homework is like, what the community is like, what parent involvement is like, etc. Try to get to the heart of the philosophy and culture of the programs. Some programs are more "rigorous"/academic and some are more developmental/progressive. I don't think it's particularly valuable to ask parent opinions as to what you should do because frankly unless parents are unhappy, all will argue the merits of their own choices. We all like to think what we are doing is best for our kids. Also, I will say that I think you will find engaged and organized parents in all the schools you mentioned.
3.) Solicit input from preschool teachers and educational assessment people (the people who gave the WPPSI). They can also give insight into the particular considerations unique to your child and family situation.
4.) After doing as much research as you can, if your child is admitted into a private - then go with your gut. It reminds me of the search for a home. Where can you envision your child/family? No matter what, there will be trade-offs for your family. Which ones can you live with?
Our DC is thriving and happy at school. I can't tell you what joy that brings me. No matter what path you choose, I think it's a really valuable exercise to look around and reflect a great deal about your family and DC's needs before starting school. Good luck.