Yes even though groups can be closed on FB, there are still rules and failing to stick to them means potentially being reported and the group being taken down
Closed means when someone posts something, you have to be in the same group as them to see it
If you make an account just for that and only join closed groups, nobody would see "John Smith Wears Diapers" just randomly on their page; unless they were in the same group
And then if it is a fake name they wont know who you actually are
It would not just be plastered on all of FB for millions to see like if you post something on a normal page that is open to anybody
Regardless I personally wouldn't show my face in pictures, just to protect yourself from worst case scenario (if someone infiltrates the group and decides to repost stuff elsewhere it could be bad, anybody can ask to join these groups and you dont know who they are or their intent)