My name is Matthew as well and my Russian visa says Мэттью so I guess that's the best variant. But don't get that tattoo. It looks ridiculous (in both languages) + no Russian will understand it.

Like why would you get a tattoo of USA in Hebrew or a German name written with oe instead of with an umlaut?

I'd advise him to get Матвей... or if you want, Матфей, though that is pretty archaic and if you Google it, you get mostly religious websites back.