If your goal is really to heal whomever the FATE boss is attacking, you can target <tt> (that's the target-of-target tag) or use the "Target the Target of your Current Target (Assist Target)" keybind (in /keybind >> Targeting).

I know your question is specifically relevant to controller players, but as you mention other players' buddy chocobos and tanking FATE bosses, it may be helpful for kb/m healers to know that the target of target health bar is a valid mouseover entity for <mo> heals.



However, if your goal is specifically to target other people's companions, for any reason, there's not much you can do besides devoting your focus to the companion in question so you can target <f> or use focus target keybinds.