Another possibility is that an evil spirit has possessed the chaos soldier. In this case, the spirit would be the main antagonist, using the chaos soldier's body to carry out its malevolent plans in the Naruto universe.
It could be that there is a mastermind behind the creation of the chaos soldier. Maybe a rogue ninja who wants to use the chaos soldier to overthrow the existing ninja villages and take over. This person would be pulling the strings and be the true antagonist while the chaos soldier is more of a tool.
Maybe Akainu from One Piece (in a crossover sense within the fanfiction). He could be warped into this Naruto - Dragon Slayer world and his extreme sense of justice could be twisted into a form of tyranny. He sees the chaos and magic in this new world as a threat that needs to be eliminated, and in his overzealousness, he becomes the main antagonist, causing destruction and pitting the ninjas and dragon slayers against him.
One idea could be the chaos soldier having a unique jutsu that disrupts the normal chakra flow in the Naruto world. Maybe it causes unexpected alliances to form as everyone tries to figure out how to counter it.
Maybe an enemy spy who has infiltrated the little soldier's unit. This spy would be constantly trying to undermine the unit's operations and would be a constant threat to the little soldier and his comrades. He could be very sneaky and hard to detect, which would make him a formidable antagonist.
It could be someone from the Naruto universe who doesn't like Naruto's newfound fascination with Aku. For example, Danzo might see Naruto's interest in Aku as a threat to the stability of the village and try to stop Naruto from further exploring Aku - related things. So, Danzo could be the main antagonist in this case.
Another possible antagonist could be a rogue demigod who is corrupted by the power of Chaos. This demigod might envy Percy's connection to Chaos and decide to turn against him. He could form his own group of followers, causing chaos (literally) in the demigod world. They would oppose Percy and his friends at every turn, believing that they are the rightful heirs to the power of Chaos.
It could be an ancient and powerful being from the chaos realm who sees Percy and Piper as intruders and a threat to its own power. This being might have been ruling over a certain part of Chaos for eons and doesn't want the demigods to disrupt the status quo.
It could be an ancient evil god who manipulates Percy into betraying the son of chaos. This god might have a long - standing grudge against the son of chaos or his lineage and sees Percy as a useful pawn.
It could be an ancient and powerful Titan who has escaped from Tartarus. This Titan wants to use Lady Chaos's power to take over the world. He sees Percy as a threat because of his past victories against the Titans and his connection to Lady Chaos.
A powerful chaos deity could be the main antagonist. This deity might see Percy as a threat to their plans of spreading chaos throughout the world. They could constantly send their minions to attack Percy and try to corrupt his mind.
A powerful primordial being awakened from the depths of Chaos could be a great antagonist. This being could have powers that surpass even the Olympian gods and a desire to reshape the world in its own chaotic image.