I think I tried this ages ago, but I can't immediately find anywhere that I documented it, so it'd probably be worth doing again just to be safe. Also, maybe getting all the dual XP adventures or having all the bird techniques or something will unlock something new here.
Anyway, my current theory about the Ring Test is that having multiple instances of as many different evasion techniques as possible is important, so if you only have one instance of Cloak of Feathers, it can't hurt to have another one. For that matter, it may turn out to be important for getting the bird avatar.
EDIT: Looking more carefully, the only mention I can find of having tried this wearing the full bird outfit was from before we knew that unequipping the cloak terminated chains, so I tired to get this adventure wearing the full outfit and only got a feather (presumably because I had broken the chain without realizing it). So it's definitely worth trying now in the full outfit.
EDIT AGAIN: Nope, full bird suit equipped, Ocean Sight active, all 4 known bird techniques in my deck, in the Dockside Sewers since that seems to be the most raven-y location, and I still just got Cloak of Feathers.