Perhaps the antagonist could be a fellow demigod who is jealous of Percy's popularity and power. This demigod starts spreading false rumors about Percy, which makes Percy really mad. So, Percy has to not only deal with his anger but also prove his innocence and stop the other demigod from causing more harm.
It could be a new character, like a powerful wizard who has a grudge against Percy for some reason. Maybe Percy accidentally disrupted one of his spells, and the wizard is out to get him, which makes Percy mad.
Another option is a group of anti - magic fanatics. They view Percy's wings as an abomination because they are a result of magic. They hunt Percy down, believing that by getting rid of him and his wings, they are protecting the world from the 'evils' of magic.
Doofenshmirtz could be the antagonist. He might create a contraption that directly or indirectly causes Phineas to get mad. For example, his -inator could accidentally zap away all of Phineas's blueprints for his latest project. Phineas would be understandably angry and would then see Doofenshmirtz as the source of his frustration.
One possible plot could be that Percy gets mad because his friends don't believe him about a new threat he has discovered. He might storm off and try to face the threat alone, leading to all sorts of dangerous situations.
It could be a new character introduced specifically for this fanfiction. Maybe a bully who has just transferred to Percy's school and takes an instant dislike to him. This bully might be jealous of Percy's popularity or has a grudge against him for no real reason other than wanting to assert dominance.
It could be a jealous rival. This rival might have always been envious of Percy's popularity and skills. So, they manipulate Percy's friends into betraying him to get Percy out of the way and take over his position.
Perhaps Percy forgets an important promise he made to Annabeth. In their relationship, promises are significant. If he fails to keep one, like a promise to help her with a project or be at a certain place at a certain time, Annabeth would surely be mad. It could also be that Percy accidentally breaks one of Annabeth's creations or possessions, which she holds dear.
It could be a new Titan that has been awakened from an ancient slumber. This Titan might be angry at the gods for imprisoning them long ago and wants revenge. Percy's death could be part of the Titan's initial plan to strike fear into the hearts of the gods and demigods, but Percy's rebirth becomes a major obstacle for the Titan.
It could be a jealous demigod. Maybe someone who has always been envious of Percy's popularity and abilities. Now that Percy has wings, this demigod sees it as an opportunity to make Percy look bad. They spread rumors that Percy's wings are a sign of him being cursed by the gods and try to turn other demigods against him while he's already feeling sick and struggling with his new wings.
It could be a jealous demigod. Maybe someone who has always been envious of Percy's popularity and power. This demigod could spread lies about Percy that lead to his disownment, making them the main antagonist.
It could be a lone wolf type of character. This person is a survivor in the same world as Clementine but has gone a bit crazy from all the hardships. They see Clementine as a competitor for resources and decides to shoot her to get rid of the competition.