Hera could be a main antagonist. Since she's often jealous and protective of Zeus, she might not be happy with Percy being adopted. She could try to undermine him at every turn. Another could be Ares. He might see Percy as a threat, especially if Percy starts to gain more power as Zeus' adopted son.
In this fanfiction, it could be that some of the older and more traditional demigods become antagonists. They might believe that Percy is getting special treatment just because he's Zeus' adopted son and not because of his own merits. So they could plot against him, trying to expose him as unworthy or even try to harm him to show that he doesn't deserve his new status.
Some of the minor gods who feel overshadowed by Zeus and Hera might be antagonists. In this fanfiction, they could see Percy as a symbol of the over - dominance of Zeus and Hera. They might plot against Percy to send a message to his parents. These minor gods could use their unique powers and their knowledge of the magical world to create all sorts of problems for Percy, from cursing him to setting up traps in his path.
Some of the more malevolent spirits or monsters from the Underworld could also be antagonists. They might resent Percy for being Hades' adopted son and try to undermine him. For example, a powerful undead general who believes he should be the one favored by Hades instead of Percy.
The Titans could still be antagonists. Even though Percy is now associated with Artemis, the Titans might see him as a threat or a tool to use against the gods. Kronos, for example, could try to turn Percy to his side by offering him power or by threatening those he cares about in Artemis's group.
The Olympian gods could be the main antagonists. Since Percy is the adopted son of Kronos, they might see him as a potential threat and try to stop him before he can fulfill any of Kronos' possible plans. They may send their demigod heroes after him.
Another god could be the antagonist. For instance, Hades might be jealous of Demeter's newfound 'son' and the attention Percy is getting. He could send his minions to cause trouble for Percy or try to undermine Demeter's influence on Percy, leading to a series of battles and challenges for Percy to overcome.
The Titans could be antagonists. Since Nyx is an ancient and powerful goddess, the Titans might see Percy as a threat because of his connection to her. They could try to either turn him against Nyx or simply eliminate him.
The Titans who were once associated with Pontus might be antagonists. They could see Percy as a threat to their plans if he is related to Pontus. They might try to stop him from fulfilling any tasks given to him by Pontus or try to use his new identity against him.
Some of the traditional sea monsters like the Kraken could be antagonists. Since Percy is now in a more prominent position in the sea world as Amphitrite's adopted son, the Kraken might see him as a threat and try to attack him.
Another possibility could be a group of rogue demigods who are against any form of association with the underworld. They view Percy's new identity as something dark and evil. They form a group to oppose Percy and try to drive him out of the demigod community. They might even try to frame him for crimes he didn't commit in order to get the other demigods to turn against him.
Another possible antagonist could be some of the more power - hungry and jealous demigods. They might not like that Percy has been adopted by Apollo and is getting special attention. They could try to sabotage his missions or spread rumors about him to turn the other demigods against him.