What does marking myself "Away" or "Busy" do?
Solution 1:
Unfortunately, this is a downside to having RealID friends. Unless it's different in D3 than it is in WoW and Starcraft, your RealID friends will always know if you're online. As you said, if you're marked as busy, people can't join a game with you. If you're away, I think it's just cosmetic to show that you're not there, so they don't think you're just ignoring them (or are you?). The best option I can see in your case is to just mark yourself as busy so it ignores group requests.
Here's Blizzard's information on the matter.