I don't see a real confrontation between Perospero and Carrot happening, and even if it does happens Perospero will certainely not be defeated given how easely he would beat normal Carrot and that I do not see Sulong Carrot as able of beating him.
Perospero has obviously a role to play during the raid of Onigashima and after, especially since he is the only Charlotte children to have arrived here so far (thanks to that BS waterfall plot) and that he seems to have made a pact with Marco.
It seems like some people here don't realize that if Oda wants Carrot to defeat Perospero he will do that because of story line purposes, now I agree that Carrot in base form should not be able to defeat Perospero however she might have a slight chance in Sulong form.
If Oda is going to have Perospero get defeated alongside Big Mom then he is probably going to have Carrot be the one to fight and defeat him whether people like it or not, Oda has already setup some possible confrontation between the two and also if they do fight there is also that possibility that maybe Carrot will have help against Perospero.
As if Pedro didn’t have either of those... Peros is many lvls ahead of Carrot using Sulong and awakening haki which obviously isn’t gonna be that strong wouldn’t change the outcome.
The only minks who surely can defeat Peros are the dukes and no one else.
Pedro, a Guardian and veteran New World "pirate", got no-diffed.
Carrot gets neg-diffed.
Sulong Carrot maybe gets low-diffed?
She needs help to take him down. Brook and Chopper got trounced by him as well. Maybe the three of them could pull it off with a nice helping of plot armor? The battlefield is extremely chaotic right now. There's no telling what kind of bad luck might befall our dear Perospero. If Pekoms were around I'd throw him in there since he's also got beef with Peros due to Pedro's death, but he's nowhere to be found right now. I wonder if he's hitched a ride with Germa or the Sun Pirates.
Marco and Peros seemed to have struck a truce or something though so we will have to see how that plays out.