Please do if you feeo like, I'm interested
Well first of all this was Pedro's thing. Pedro left Zou originally to find Poneglypha and that's how he ended up at WCI and got his lifespan stolen
Then with the straw hats he dedicated his last mission to helping Brook find and retrieve the poneglyph rubbings.
If Carrot is supposed to do something in memory of Pedro, then fighting perospero isn't it. It's following in his foot steps. Finding a poneglyph.
Also there's an entire Japanese Folklore tale about this believe it or not. It's a story explaining why there seems to be an image of a Rabbit on the moon.
There's a story about A Rabbit, A monkey and a Fox. The monkey being Luffy, the Fox being Kin'emon and the rabbit being Carrot. In the end, The Rabbit goes through some shit but gets rewarded with its Image being permanently engraved on the moon.
I think "permanently Engraving on the moon" represents Poneglyphs since only Poneglyphs can be "Permanently engraved" in One piece. And the moon connects to Sulong and the Kozuki clan.
In end there's both a story reason and outhe of story reason why Carrot's main contribution to the arc will be finding the road poneglyph.