Is there a record of species kept in zoos worldwide and number of individuals? If yes, can one have access to it?
A potential data source could the data base that stores zoo animal data, where the most comprehensive (to my knowledge) is Species360 ZIMS (Zoological Information Management System). It contains animal records and current breeding programmes, but also information on how to care for animals (best-practices and studbooks).

In your case, the ZIMS for Studbooks module should be relevant, which contains:


  ZIMS for Studbooks improves Studbook Keeper efficiency and record accuracy with suggested animals, notification of records updates, and extensive data quality tools. [...] enabling users to conduct intensive population analysis, improving species and conservation management


However, I do not have access to the data base myself, so I cannot say how accessible the data is. If you are connected to a zoo, teaching institution or research facility you can however apply for access and a membership, but I don't know the specifics of this process (or possible costs). It might be worth checking out either way.