This is likely due to:
* His family's notoriety: The Getty family is a famous and wealthy dynasty, and Balthazar's full name helps differentiate him from other members of the family.
* His name's unique sound: Balthazar is a distinctive name, and using the full name makes it easier to identify him.
It's possible that within his family or close circle of friends, he may have a nickname, but this information is not publicly available.