The teachers could be antagonists. Since Percy is a rebel, they would be the ones trying to enforce the rules that he's breaking. They might give him detention, assign extra homework as punishment, or constantly be on his case.
The setting of a high school is important. In 'percy is a rebel highschool fanfic', the high school environment would likely have strict rules that Percy rebels against. This could include things like a dress code or a curfew on campus. Also, the different cliques in high school could play a role. Percy might not fit into any of the traditional cliques and instead forms his own with other outcasts. And the plot might revolve around how he deals with the consequences of his rebellious actions, like getting in trouble with the principal or facing expulsion.
The Titans could be antagonists. Even though they have been defeated before, some of them might hold a grudge against Percy and Annabeth and see their baby as a new target for revenge. They could try to kidnap the baby to use it as a bargaining chip or to corrupt it to their side.
A strict teacher could be an antagonist. Maybe this teacher has old - fashioned views and doesn't like Percy's unorthodox ways or Artemis's independent streak. The teacher could constantly give them detention or try to stop them from participating in certain school events that are important for the plot.
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 in a 'Percy back in time fanfic' could be the followers of the Titans. They were numerous and loyal to Kronos. They would view Percy as an enemy since he is from a time when the Olympians are in power. Additionally, if Percy travels to a time when a powerful and evil demigod once lived, that demigod could be an antagonist. For example, if he goes back to a time when a demigod was on a path of destruction before being stopped, Percy's arrival might disrupt the events that were supposed to stop that demigod, and so the demigod would turn against Percy. There's also the possibility of some ancient sea monsters that were more prevalent in the past and were under the control of the Titans or other evil forces.
The Titans could still be antagonists. Since Percy has a connection to Hecate, they might see him as a threat to their plans of overthrowing the gods. They could try to capture him to use his powers for their own evil purposes.
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.