Hello!
My 2 cents. While my then DS8 was already identified as gifted, I think I understood I needed more information about him. However, that information was going to be used to decide whether to send him to the gifted magnet, which your DS is already in. It was a tough decision though for us because we were considering transferring him from our high-performing flashy school down the street where he'd been for 3 years to the gifted magnet. The results on the private IQ testing convinced us that we needed the magnet for him. 3 years later it's clear it was the best decision for him and us. So bottom line is that private testing gave us a wealth of information that we did not have from the OLSAT.
Good luck!