I've thought about this in the past, believe it or not. I'm not so confident with the Imu and Blackbeard associations above, but I am here. I really struggle to see the Kouzuki clan as being narratively representative of the moon, even if moon is in their name and in the rebels' tattoo.
I question that association because
(1) Five of the
known samurai families also have 月 in their name.
(2) It was
Orochi who was represented by the moon in Toki's prophecy.
(3) Momo is prophesied to lead everyone through the
dawn.
(4) The Kouzuki crest has a
sun in it instead of a moon.
(5) The crescent moon tattoo is even
waning. (Interestingly, Momo does have that waning crescent moon pattern in his imperial robes but his Kouzuki clan crest is much bigger.)
So to me it feels like the name doesn't matter (at least outside of Wano) and the waning crescent moon tattoo is more about Toki's prophecy-- shorthand for a fading moon. (Toki's Amatsuki family crest also has a waning crescent moon with a bird in it, I just noticed when counting families with 月.)
Am I missing something else that better supports the narrative weight of the moon being associated with the Kouzuki clan?