Since your development environment evidently enforces foreign key relationships, changing the value of a key is a problem.
Designing the database so that an important fact like SSN isn't an internal key in the first place (too late to do that).
(If SSN in your database doesn't represent an important fact, then it may not be necessary to change it in the development copy, since it only exposes meaningful data when connected to the real personal data in production, which you are removing. But you and your colleagues may disagree with me.)
Changing data for one SSN at a time - for instance input a row of 000, Proxy, Proxy in TableA; change all SSN of 111 in other tables to 000; change SSN 111 in TableA to 999; change all SSN 000 in other tables to 999. I have done this in production data, when someone was recorded with two different service IDs in different databases and we needed to merge together the data from two IDs.
Copying data to a database that does not have referential constraints on the tables; change the data however you like, but finishing with correct relationships; copy the data again into another database where there are constraints.
If the language and design allows, delete or turn off the foreign key enforcement, change your data, then create or turn on the foreign keys. But beware of constraints in the database that were already disabled. If at the end you turn on all constraints that were disabled, you may also turn on the constraints that were already disabled before you started.
Standard tools to produce anonymous identity data may be useful to you. For names, I've used a method of taking a list of reasonable and distinct forenames, and a list of surnames, two different prime numbers in length (or "co-prime"). Take names in (semi random) order from one list, and from the other, in turn, returning from end of list to the start. Combinations of names shouldn't be repeated until all pairs have been used once. My list included one spelling of "Mohamed", and as a surname, because I thought it would be provocative to produce a "Mohamed Singh". It also didn't occur to me to use "hsggghhgh", but you could use "User101first" and "User101last", etc.