I voted for the Muni for a handful of reasons. I am a golfer, and while I absolutely love getting to a country club maybe once a year, I don't know if it's really where I want to be every day. Some of the people are just waaaay to stuffy. Side note - this is a job I think I might be interested in doing long-term as you plan to, so as you said FIRE wouldn't be as much of a priority.
I also know someone who worked at a club for many years and it was what you described with a silly amount of hours, and a pecking order. I think the only category where the CC wins out is location; only 5 miles from home. If that's what's most important, make it your deciding factor, otherwise go public.
I like the public because you have a higher pay, a pension, a better spot in the pecking order, a 40 hour work week, and a full 12 months of pay. I understand it's not as prestigious, but it seems like the smarter choice.
My only other thought is if you like to travel or have other strong interests, maybe the 3 months/year off work would be a better reason to go Private.