Maybe I don't understand where you're seeing the Real ID name, but it sounds like you've added yourself to your own friends list? If so, why can't you just add yourself to your friends list with the BattleTag instead of Real ID, to avoid showing your name altogether? I don't think there's a single benefit to using Real ID over BattleTag, is there? I dropped Real ID years ago when BTags became a thing for WoW because: A) like you, I didn't want my full name showing up everywhere, and B) other peoples' names were showing up in my chat window as I was recording videos. So, now the only thing that shows up on other peoples' game clients is "BTagName has come online."

I've also disabled Real ID in my account, so perhaps that's what you're looking to do, instead.