My wife and I bought two copies of FFXIV - one outside of steam and one through steam (fully steam integrated). She started a few weeks before I did. When we added FFXIV to steam, we had massive problems getting the game to launch. We kept getting the dreaded "Unable to complete version check" message to the point we couldn't even launch the game at all, it was fully stuck. We took FFXIV off of steam completely and it works just fine. Now I just launch the standalone game that my wife purchased and I simply just put my login information into the launcher and I can play it without any problem outside of steam.

Now I apologize that I can't answer your question directly about whether you can get FFXIV fully integrated into steam, just be warned that having it so was only massive headaches for me.