I'll also second the advice to get a private eval - getting your ds into the gifted program is one goal, but there's another piece of the puzzle that needs to be addressed - it sounds like your ds may have a very real disability that he will need help with and accommodations for. I also thought of selective mutism. FWIW, I have a ds who has an expressive language disorder - although my ds' expressive language disorder primarily impacts his written expression, it also pops up in not-so-obvious places in verbal communication too. He struggles a TON with talking to adults, particularly teachers at school. He's working with an SLP and it's helped him soooo so very much.
Having a diagnosis, if there is one, is worthwhile for several reasons. It's going to give you something you can use to advocate for accommodations and also use to advocate for a different ability measure to be used for the gifted program admission. It's going to help you determine how to get effective help for your ds, and it's going to help your ds to know for himself, and to be able to have the adults he's spending his time with at school, know that it's not just him being silent purposely, or something "wrong" with him (kid-think), it's something that is different and can be helped.
Best wishes,
polarbear