As an American named "Joseph", I went by José (Spanish version) in elementary school and يوسف / Yusuf (Arabic version) when I lived in Egypt. In the military, I was "Peterson" or "Sir". Some people call me "Joe" even though I never refer to myself that way, but I don't insist on a name. Names aren't for me; they're for other people's convenience.

You can have your cake and eat it too. When you introduce yourself to someone, give them both options -- the authentic Mijail and an easier Americanized version such as Mike. Some Americans will struggle with the pronunciation of Mijail and opt for the simpler, more familiar substitute. Other Americans will enjoy the opportunity to learn and adapt. They can try Mijail.

I see no reason to rigidly insist on a single name. Nor do I see any need to abandon your culture here in the USA. Introduce yourself with a two-fold name, as a choice, depending on the other person's comfort level.