There is a major problem with group invites and I’ve noticed that my status is set to Busy. Setting it to Available holds as long as the list is open but as soon as I close it it reverts to Busy.
I’ve tried to relog, even restart BattleNet but no avail. Funny thing is that as long the list (set to Available) is open I can invite people but then the Start Game is grayed out. Making the invitee to leader solves the latter, i.e. we are in group and can play but my status still reads Busy.
Cheers Euclid
Edit: I did run Scan and Repair, did not resolve the issue. E.
I’ve been having the same issue for a while now.
I have the Blizzard Desktop App set to exit completely when I start a game.
If I go AFK, I’ll bring up the Friends Menu in-game and set my status to “Away”. When I come back, I set my status to “Available” and close the Friends Menu. If I open the Friends Menu again, my status is reset to “Away”.
I have been able to work around this by tabbing out of the game. I then run the Desktop App. In the upper right corner, next to my username, I click the drop-down menu and set my status (in the App) to “Online”. I close the Desktop App and maximize the game. When I open the Friends Menu I find my status set to “Available”.
I hope Blizzard can fix this soon. 
Yes, that works if you are on your “home server”; mine is EU. Tested it with US and Asia server and there it won’t work. Cannot invite there because the status keeps resetting to Busy 
Cheers Euclid
Edit:
Found a workaround:
1# select your character
2# select torment level
3# open friend list
4# set Available
5# while list is still open, enter the game, it will remain open
6# invite from list
7# once all members are in the game, close list
Cheers Euclid