A believable character arc for Percy could involve him initially struggling with the negative connotations of being associated with Arachne. But as he delves deeper, he realizes that he can use his new status to bring about change. He could learn from the Arachne descendants' resilience and use that to become a stronger leader among the demigods. He might start off as an outcast because of this new identity but end up being respected for his unique perspective and abilities.
First, start with Percy being in denial about his new identity as the son of Arachne. Then gradually have him accept it as he discovers the good aspects of this heritage. For example, the power and loyalty of the Arachne - related demigods.