I've made jokes about one of the kids attempting to domesticate the Bats but I see Damian finding an injured bat in the Cave and Damian being Damian immediately brings the little fella to Alfred who agrees to look it over but he's not a vet and there isn't a lot he can do. Damian begs Bruce to take the Bat to the local vet but Bruce is wary because one, he doesn't want Damian getting attached in case the bat has to be put down or worse, it gets better and he bullies Bruce into letting him keep it and two, he's technically meant to be famously afraid of bats. He can't just rock up to the vet with a bat because he can't risk a) a vet getting curious about the bats on the estate b) blowing his cover as the guy so terrified of bats that there is no possibility that he can be Batman. Also if Damian gets a bat all the kids will want one. But Damian cracks put the puppy eyes.
Bruce shows up to the smallest local vet practice in the city, dressed in a hat, shades and his civvies with the bat on a carrier. He gives a false name, uses a fake card but the receptionist recognises him and the story leaks that Bruce Wayne is finally over his fear of bats - or working through it - and he's adopted a little bat. A News Channel tries to get him to meet Batman on camera which Bruce declines as it is too big of a step.
The bat does become a pet of the family, he damaged his wing too much to fly very well so the little guy just chills on the Bats and Robins when they're around, sometimes sits on Alfred's shoulder or hangs from the little railing he had installed in the kitchen. Bruce is still wary of the little guy but will pet him or feed him some dried insects when he's feeling up to it.
