That’s a pretty neat idea, anon, though I don’t think she’d have to wait until ANBU to get that title (and having a title as an ANBU that isn’t your animal codename might be a bit counterproductive anyway?)
Because if you consider it? Sakako (maybe) doesn’t have the hawk contract, but she does have an emergency summon and combining that with the idea that she can manifest people’s angry ghosts AND that Tengu (according to wikipedia, at least) “possess people, especially women and girls, and speak through them,” to a very superstitious person and in the right series of events she would look like a Tengu.
Especially if it’s… let’s see… a business mogul or corrupt noble who is going to deforest/mine a place that’s sacred to the local village. And the local village hired Next Gen Team Seven for a really small task (maybe help clean the shrine up on the mountain, or escort the village chief who is old up the mountain so he can clean the shrine, etc. etc.) but then business mogul / corrupt mogul shows up and does *typically upper class awful things* and obviously any Team Seven wouldn’t let this happen so they kind of stay and fight and… well.
What Team Seven doesn’t have a flare for the dramatic?
Sakako uses a genjutsu on the bad guy, brings their ghosts to life (so to speak) and maybe has to summon her dad’s hawk when she gets pushed off a cliff and it’s all just. Bad guy has had a religious experience and learned the error of his ways and, hey, the local village can actually help business person/noble how to mine/harvest whatever they were trying to get in the first place in a sustainable way without razing everything to the ground (and that brings new blood back to the village which was, admittedly, beginning to age out–no one in that village was younger than forty, so there wasn’t much future for the village).
… so maybe it’s not so much her title as it is a significant note that would be put on her bingo book page (“actually possessed by a tengu? can summon tengu and ghosts?” Poor Rock nin are so damned confused.)
For such a mythological nickname, I figured a dramatic incident conflated to an incredible degree would work pretty well. With Shikako’s Shikabane-hime, it was after her badass fight against Gaara–one time to make a massive first impression connected to an intimidating but easy to remember moniker.
Tengu are more connected to birds of prey than stags, so her connection to hawks via Sasuke is probably more relevant than her Nara heritage. And it’s believed that the long nose is reminiscent of a beak so a bird ANBU mask would be enough, I think, and I’m not sure ANBU masks are overly detailed?
I kind of like the idea that she begins collecting fearsome nicknames early on like her mum–ANBU has a bit of a sense of anonymity that I’m not sure will transfer over well? Like, when Kakashi was ANBU he was well-known as an elite ANBU and all the enemies knew to beware of the ANBU with a wolf/hound mask, right? But I don’t think they’d go as far as to call him something else like Raikou. They’d just call him Wolf. Likewise, they’d probably just call ANBU Sakako Hawk or Falcon or whatever mask she’s assigned.
But I guess a way that could work is if… hm… if Sakako inherits Sasuke’s ANBU mask which is Hawk/Falcon, she’d be stepping into this role with a lengthy and no doubt prolific career attached. And enemies are already fearful of Hawk/Falcon but they’ve notice that Hawk/Falcon’s fighting style/height/whatever has changed slightly and this is a new Hawk/Falcon. So to differentiate the two they call the first one (Sasuke) “Garuda Hawk” and the second one (Sakako) “Tengu Hawk” and then later just drop the Hawk part.
Or, well, maybe Sasuke doesn’t so much pass down his mask as he does part-time ANBU work and their masks are very similar… so still the same reasoning just now the enemies aren’t always sure which Hawk they’re dealing with (surprise, it’s both of them this time! Badass father-daughter ANBU bonding!)