Some of the old monsters from Greek mythology that have a vendetta against Aphrodite's lineage could be the main antagonists. Maybe a spurned former lover of Aphrodite who was turned into a monster. This monster targets the daughter of Aphrodite and Percy because they are symbols of Aphrodite's power and influence. It constantly tries to harm them and thwart their plans.
There could be a new, dark force that is emerging in the world of demigods. This force is threatened by the unique combination of Percy's strength and the daughter of Aphrodite's powers. So, it sets out to destroy them. It could be an entity that feeds on negative emotions and wants to plunge the world of demigods into chaos by getting rid of those who can promote love and unity, like the daughter of Aphrodite and Percy.
The children of Ares might also act as antagonists. They are often at odds with the followers of Aphrodite in the Percy Jackson universe. They might see Percy being the champion of Aphrodite as a threat to their own power and influence. So, they could constantly try to pick fights with Percy, both physically and by trying to turn other demigods against him.
The Titans could still be antagonists. In the Percy Jackson universe, they are powerful and malevolent. A princess in a fanfic might have to face their wrath as they try to overthrow the gods and take over the world. They would see the princess as a threat, especially if she is allied with the gods or has some special power that could stop them.
The main antagonists could be some of the other gods who are jealous of Aphrodite's special situation with Percy. For example, Hera might not like the idea of Aphrodite having a child with a demigod, as it goes against the normal order of things in the godly realm.
Percy is a must - have main character. As for Aphrodite, she's central to the concept of the trap. Another character that could play an important role is Grover. Grover could sense that Percy is in danger and try to find a way to rescue him. His satyr abilities might come in handy in detecting the magical aspects of Aphrodite's trap. And of course, we can't forget about the minor gods or goddesses that Aphrodite might enlist to help with her trap, which could add more complexity to the story.
The Titans could be great antagonists. They might see Percy as a threat because of his connection to Thanatos and the power it could potentially give him. Also, some rogue underworld creatures, like powerful demons or vengeful spirits, could oppose Percy. They may not want Thanatos to have a more powerful influence through Percy.
Some of the more powerful and vengeful spirits in the Underworld itself could be antagonists. They might not want an outsider like Percy to be the Champion of Hades. They could try to undermine him at every turn, like spreading false rumors about him among the other Underworld denizens or directly attacking him when he's vulnerable during his quests in the Underworld.
Some of the primordial gods who are jealous of Chaos's newfound relationship with Percy could be antagonists. They might view Percy as an upstart and try to undermine his position. For example, Erebus could try to create chaos in Percy's life in a more malicious way, not in the way that Chaos intends.
The Titans could still be a major threat. Even though they were mostly defeated in the original series, some of their followers might see Percy's new role in Atlantis as a threat to their plans for chaos. They could try to undermine Percy's position in Atlantis and use the underwater city as a base for a new uprising.
It's possible that the main antagonist is a new mythological creature that has the ability to manipulate minds. It has lured Percy and other demigods to the mental hospital and is slowly driving them insane in order to feed off their fear and chaos energy.
The Titans could still be antagonists. They might see Percy as a threat in his new role as the champion of Hecate. Another could be some rogue demigods who are jealous of Percy's new power and want it for themselves. And there could be magical creatures that have been corrupted by dark magic and oppose Percy's efforts to bring balance with Hecate's magic.