I would just let him do what he wants, OP.
My multi-ethnic children were given 3 first names, one for each of our cultures. My son has always gone by his third first name, because we all think it really fits him. My daughter has always gone by her first, English-friendly, first name. But in middle school she suddenly started telling everyone her name was Elizabeth. Now in high school, some kids still believe her real name is Elizabeth

She thinks it's funny. I couldn't care less.
I understand why you feel your culture is rejected, but please understand that your child is in the prime years to want to fit in, and when he's older, he will treasure his origins much more. It's a known developmental progress in multi-cultural children. What you need to do is enroll him in native language classes, feed him movies and books in that language, cook him all the great food of your country, and make his stays pleasant there if ever you return there for visits. Point out repeatedly that being multilingual makes you smarter, and is considered an asset as an adult. It looks great for college admissions too!