What is the adopted name for a newborn girl if her name at the time of the 'aqiqah was "Badi'ah" and her name in official documents is "Fatimah"? And is it permissible to call her by either of the two names?

There is no objection to naming a girl Badee‘ah, nor is there any objection to changing the name to a better one, such as Fatimah. The fact that the new name is better does not preclude calling by the old name, and there is no objection for a person to have two names.