A group of rogue demigods who have been influenced by the dark side. They envy the power that Harry has as a wizard and want to use his magic for their own evil purposes. They would constantly be creating trouble for Harry and Percy, trying to break their bond and steal their powers.
A powerful dark wizard who has made a pact with a Titan. This antagonist could be using dark magic to corrupt the boundaries between the wizarding world and the world of the Greek gods. He wants to gain ultimate power by controlling both realms, and Harry and Percy would have to stop him.
The Giants might also be antagonists. They could band together and target Percy's kingdom, believing that by destroying him and his rule, they can undermine the power of the gods further. They might use their brute strength and strategic thinking to attack Percy from different fronts.
Another option could be a group of rogue demigods who have turned to the dark side. They follow Percy to Forks, hoping to find some powerful artifacts there. They don't care about the vampires or werewolves and are willing to cause destruction to get what they want. Percy has to stop them while also dealing with the local supernatural creatures.
In this fanfiction, Voldemort or his remaining followers could be major antagonists. They wouldn't like the idea of Harry having new allies in Percy and the demigod community. From the Percy Jackson side, the Titans or some of the more malevolent gods might not be happy with the alliance either. They could try to break it up or cause trouble for the two boys, seeing them as a potential force against their own plans.
In some fanfictions, Voldemort or his remaining Death Eaters could still be the main antagonists. They might target Harry and Neville specifically because they see them as a threat to their plans of taking over the wizarding world again.
It could be a rogue Death Eater who escaped the clutches of justice after Voldemort's fall. This Death Eater might be seeking revenge on Harry and decides to target Bill as well since he's Harry's mate.
The Titans could be the main antagonists. Even though they have been defeated before, some of them might have found a way to return or their followers could be causing trouble. They see Artemis as a threat or a target for their revenge against the gods. Another possibility is a new group of powerful monsters that have banded together and are specifically targeting Artemis for her power and influence among the demigods.
Well, the main antagonists could be a group of rogue demigods who have formed a cult around Kronos. They believe that Percy, as the heir of Kronos, should lead them in overthrowing the current order of the demigod world. But Percy doesn't want any part of it, so they become his enemies. Another possible antagonist could be an ancient being that was once in league with Kronos and now sees Percy as a means to resurrect Kronos or continue his dark legacy. This being would stop at nothing to force Percy to embrace his role as the heir.
Perhaps some of the older, more traditional gods could be antagonists. They might see Percy as a threat because he's now the God of Dinosaurs and they fear the power he could gain from controlling these ancient and powerful creatures. For example, Zeus might be worried that Percy's new role could disrupt the existing power balance among the gods.
The Roman gods could be antagonists. Since Percy is a Greek demigod, they might not fully trust him as the King of Rome. They could try to undermine his rule, believing that he will favor the Greek side in any disputes or decisions regarding the demigod world.
Some of the more powerful demons in Hell itself could oppose Percy. They might be jealous of his sudden rise to power as the Prince of Hell. These demons could have their own agendas and territories in Hell that they don't want Percy interfering with. They could use their dark magic and legions of lesser demons to try and get rid of him.