CT isn't an easy fix to this battle. Pain would have to go straight for it, and he doesn't do that unless he thinks he's in trouble, since it drains Nagato's chakra significantly. But if he's in trouble, KCM Naruto is already whooping him.
Also, if Naruto has the intel of Nagato being the one controlling the Pain Paths, this battle ends very quickly, because Naruto will just do the same trick with the chakra receivers that he did to find Nagato (he still has sensing abilities). Nagato has to shut down the other Paths to use CT anyway, so either Naruto uses the less threatening Paths to locate Nagato neg diff, or Nagato shuts down the other Pain Paths, and KCM Naruto goes straight for Deva Path.
That said, I don't think KCM Naruto can counter CT on his own, because Nagato claimed that he could have made a CT big enough to contain even KN8, and KCM Naruto was still afraid of CT during the fight with Edo Nagato.
As far as Deva's other options go,
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. Could KCM Naruto pull off the same feat?