Hi and welcome! Different families have had different results with approaching schools ahead of time. I am a believer in giving and getting as much information as you can ahead of time, though.
A couple of thoughts. Private schools a lot of times mean less flexibility, as they have more freedom to do whatever they want. But of course this depends on the particular school.
Is there a GT coordinator at the school? It may be a good idea to reach out to that person first to see what sorts of options may be available. Also, if there's more than one kindy classroom, the GT coordinator might be able to recommend a particular teacher that would work well with your kiddo. If no GT coordinator, the principal may be a better person to contact, unless you already have a relationship with the teacher from past kid experience. (The principal signs off on placement decisions.)
You must try to tread lightly, though, because teachers will have heard many, many stories of "my child is already reading", and they may say "we have lots of kids like that, and we know what to do." Be specific about what your kiddo is reading, and how quickly they progressed, because it's likely they'll be at a completely different level in the fall. I would suggest approaching it with "Can you help me? I'm not sure what to do - this is what my son is doing now."
Good luck!
