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AKA "inspired by that one post about Danny being a flerken and living with the Batfam" idea! Except he looks pants-shitting, "oh dear god, what is that" terrifying.
I just love slightly feral animal-like Danny in a... shape. It's not immediately identifiable as a cat or dog, maybe he has a few too many legs that kind of look like a tail at one point? And when he skitters up walls like a particularly small dog-sized tarantula, it's terrifying enough to make seasoned criminals squeal.
Let's imagine Danny had some sort of accident with a portal and was Wizard-of-Oz'd into Gotham, a literal hellmouth of a city with so many curses that it'd make John Constantine start to sweat. And this city also has... weird Ecto. (In my brain, there's a connection between the Lazarus Pit and ectoplasm, like pit waters are the sewers of ectoplasm or something.) It's enough for Danny to still exist but he can't seem to stay human-shaped. It's better than being a Blob Ghost, but not by much. His fur-scales-feathers-skin-something look dark as the midnight sky.
And who should stumble on this weird-looking Thing aside from Damian, secret animal-whisperer and passionate Pokemon collector? Damian, who known what a scared feral animal looks like and who can coax it into his arms? It doesn't matter that Danny has maybe five or six limbs. He can make himself slightly smaller at will (not in a Magical-Girl-Transformation way, mind you. When he changes shape, there's the distinct snap of bones breaking and wet, fleshy sounds of his organs, muscles, ligaments, tendons, everything shifting).
Damian has literally been trained by the League of Assassins under the Demon Head. He's likely seen more people's insides than an ER surgeon; he's killed more than enough people in incredibly grotesque and violent ways to be totally unphased by Danny changing shapes. Maybe he'll actually be sort of touched, a bit pleased, that his new Thing pet would change itself so violently so Damian could hold it.
What would Damian name it? He's outwardly violent and aggressive towards others, but pretty passionate and heartfelt once he cares for someone. Alfred the Cat comes to mind. So maybe Damian takes one look at this supposedly scary Thing and thinks, "It looks like Father."
As in, Dark as Night? A shadow inspiring fear amongst criminals? Spoken about in whispers, sometimes laughed off as a joke but still cautiously reverent, just in case?
Danny's new name is Batman.
Of course, this causes some confusion when Damian comes home to Wayne Manor and says, "Batman and I will retire to my room." In front of Bruce, who naturally and kind-of-correctly assumes his son picked up another animal while on patrol. Bruce had a hard time explaining this to a very concerned Dick, who was holding up a wooden stake and a bible (Dick totally wasn't going to kill Bruce if he turned out to be a vampire but it's always good to be prepared!), after Damian apparently made a wayward comment that "Batman refuses to eat anything besides raw meat."
And Danny is having a great time!! Sure, Damian treats him like a pet, but he gets affectionate pats on the head, incredibly expensive steak, and a soft place to sleep. He awkwardly dragged several blankets from the living room to Damian's room to make a bed in the kid's closet. (Alfred watched from behind the couch as this six-legged hairy-ish catlike Thing determinedly waddled with three blankets in its mouth, occasionally tripping on its own legs. He went back to dusting the crown moulding silently. So, that's why Master Damian requested uncooked sirloin steak twelve times in last few days. Hm.)
So, the Batfam accept there is another Batman in the family. Except they haven't actually seen Danny (aside from Alfred and Damian).
Until Dick needs to talk to Damian and goes into the boy's room. But it's empty?? He could've sworn he heard somebody talking or something in here, but maybe not? He turns to leave and then hears it again: a soft kind of thump coming from Damian's armoire. A shit-eating grin spreads across his face as his Older Brother Instincts kick in. Jason used to hide in closets and try to scare Dick when he was little; Damian, despite being a child soldier and trained assassin, was still a little kid at heart, right? The kid's clearly hiding from Dick to scare him or something.
(Damian was in the Batcave, studiously typing "Google, what non-Earth animals reside in Gotham, please?" into the Batcomputer. I like to think that Damian uses the internet like a 85-year old man who thinks a Google employee personally replies to each question.)
So, Dick creeps forward and abruptly slams open the armoire doors!! Only to let out an unholy shriek of terror as Danny, who was taking a nap, frantically skitters out of the closet looking like a Frankenstein cat-dog with bat wings. He crawls under Damian's bed as Dick scrambles into the hallway.
The cat-dog-Thing is out of the bag now. Damian looks utterly deadpan as he explains that Batman is his pet and not to concern themselves with it; Bruce, Tim, Jason, and a white-faced Dick disagreed. They need to see it to make sure the Thing won't harm anybody, especially considering it's fucking living with them!! How do they know it won't try to eat them in their sleep??
"Batman does not eat raw human meat, Todd. Why are you concerned now? It has resided with us for two months now."
"Two months?" Dick nearly faints (again).
"Yes, Batman is very well-behaved, Master Dick." Alfred, who's been feeding Danny for the last two months and has seen all the little quirks the Thing has, offers a consoling half-smile.
Ultimately, the Batfam decide to keep Batman in exchange for scary dog privileges. They'll have to think of another name for Danny considering having two Batmans in Gotham would be pretty confusing (especially if one of them decided they did, in fact, like raw human flesh).