The Titans could be the main antagonists. Even though they were defeated before, they might see Percy's rise to the throne as a chance to regain power. Kronos could have some remaining essence that tries to possess someone close to Percy to undermine his rule.
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.
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.
The Titans could still be a major threat. They might see the Beast King as a new power to either control or destroy, and they would oppose Percy's attempts to make peace with the beast kingdom.
The old - guard Olympians could be the main antagonists. They might see Percy as a threat now that he's the King of the Titans. They're used to being in power and might not want to share or give up any of their influence.
Some of the more traditional Spartan nobles could be antagonists. They don't like the idea of an outsider like Percy being king. They might plot to overthrow him and restore the old Spartan monarchy without his more modern and demigod - influenced ideas. They could use their influence over the military and the common people to try and get rid of Percy.
It could be a group of rogue demigods who want to use the Dragon King's power for their own evil purposes. They might be jealous of Percy's status and see the Dragon King as a means to gain more power and control in the mythical world.
One possible main antagonist could be a corrupted demigod. Maybe someone who was once Percy's friend but became power - hungry and turned against him when Percy became a hero king. This demigod could form an alliance with some of the more malevolent creatures from the Underworld, like the Furies. Another antagonist could be a powerful sorcerer who uses dark magic to try and undermine Percy's rule. The sorcerer could be from a long - forgotten line of magic - users in the Greek world who have a vendetta against the demigods.
The traditional gods who don't want to see Percy as king could be antagonists. For example, Zeus might see Percy as a threat to his power and try to stop him at every turn.
A group of hunters who have a vendetta against wolves could also be antagonists. They might be descendants of some ancient hunters who were cursed by a god to always hunt wolves. Percy, as the king of wolves, has to protect his pack from them.
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.