Question :
I really don’t want to go in to too much detail as to WHY I want to do this. Just know that it would be ideal for my use case.
Is there a way to disable a specific database role from using the Copy, Copy with Headers, Save Results As..., Print... functionality in the results window in SSMS?
Answer :
No, there is no way to handle this from the database/instance, unless you just deny them access to the database at all.
SSMS is a client-side tool, and the things that it can do with the data you’ve already let it consume are outside of server-side controls. Even if you could take those menu items away, there is always the PrintScreen button.