Okay, I have had some luck. If I change a trust_level group name and then go to /sidekiq and run Jobs::EnsureDbConsistency that changes the slug for me. 
I will continue testing it out to see if I can do it consistently.
Update:
I can do this consistently. And the new slugs show up in the Flair menu:
Does this work for you too? My initial confusion was obviously based on impatience. 
I think if I had left my changes in place until that sidekiq job had run as part of its regular schedule then it would have all happened automagically…