One possible antagonist could be Orochimaru. He has always been interested in Naruto for his Nine - Tails power, so it would make sense for him to be a threat that Kakashi has to protect Naruto from.
One possible antagonist could be the Akatsuki members. They are always looking for strong opponents or valuable information. Maybe they target Iruka because they think he has some knowledge about Naruto that they can use, and Kakashi has to stop them.
In this fanfiction, it's possible that an ancient and powerful beast is the antagonist. The beast might have captured Pyraa to lure Naruto into a trap. Naruto will need to use all of his skills, including his Sage Mode if necessary, to defeat this beast and save Pyraa.
Orochimaru could still be an antagonist. He might see Naruto as a threat especially if Kakashi is training him intensively. His goal of obtaining power could lead him to target Naruto.
One possible antagonist could be Orochimaru. He might be interested in Kim's unique skills and gadgets for his own evil experiments. He could set a trap for her and Naruto would have to save her from Orochimaru's clutches. Another could be Drakken. He might team up with some ninja villains, like the Akatsuki members, in order to achieve his own goals. Naruto and Kim would then have to face this combined force.
Well, it could be that in the fanfic, Kakashi saves Naruto from a trap set by Orochimaru. Kakashi arrives just in time, using his lightning - style jutsu to break through Orochimaru's defenses and snatch Naruto away from the clutches of danger. This would create a tense and exciting moment in the story.
It could be some powerful demon lords in hell. They might see Naruto as an intruder and try to stop him from whatever his goal is.
Perhaps he thinks fanfic undermines the integrity of the original Naruto story. The official story has a certain narrative and character development, and fanfic can deviate from that significantly. When it comes to his connection with Naruto, any deviation in fanfic might be seen as a negative. For example, if a fanfic changes the way he mentors Naruto, it could be something he wouldn't like.
Danzo might be an antagonist. He's always been power - hungry and might see Naruto as a threat, especially if Kakashi is guiding and protecting him. Danzo could try to undermine Kakashi's efforts to raise Naruto and might even send his root agents to spy on them or cause trouble.
The village elders could be antagonists. They might see Naruto and Kakashi as threats to their traditional way of ruling the village and use false accusations to banish them.
It could also be due to a misunderstanding. Kakashi might be keeping some secrets from Naruto, like information about his parents or some hidden power within him. Naruto, being curious and eager for answers, could misinterpret Kakashi's motives for secrecy as a sign of distrust or dislike, which then turns into hatred.