Naruto might be initially very disappointed. He has always looked up to Kakashi as a strong and cool ninja, and being refused by him would be a big blow. He could become a bit sulky and withdrawn, feeling like he's not good enough.
He could also become more determined. Naruto has a strong will. Instead of just accepting the refusal, he might try to prove himself to Kakashi. He could start training on his own even harder, showing Kakashi that he has the potential and the drive to be trained by him.
Naruto might be initially very disappointed. He has always looked up to Jiraiya and expected to be trained by him. So, when faced with the refusal, he could be heartbroken and feel like he has lost a great opportunity.
Kakashi might be initially shocked. After all, Naruto has always been his student and the sudden refusal to use the term'sensei' would be unexpected.
Eventually, Naruto could grow resentful. If the ignoring persists for a long time, he might start to think that Kakashi doesn't care about him at all. This could change Naruto's attitude towards Kakashi and might even make him more rebellious in his training or actions. However, deep down, he still probably hopes for Kakashi's recognition and approval.
Naruto would be devastated. He sees Kakashi as a mentor and father figure. He might initially go into denial, not believing that Kakashi would actually abandon him.
Well, Kakashi could train Naruto in combat skills specific to ANBU missions. This would include more lethal and precise attacks. He might also teach Naruto how to use different types of weapons that are common in ANBU operations. Maybe Kakashi would set up mock battles where Naruto has to face multiple opponents at once, just like in real ANBU scenarios.
Naruto might initially be very disappointed. He has always been eager to train and become stronger. So being denied by Kakashi could make him feel a bit down. But he could also be defiant. He might start training on his own, determined to show Kakashi that he doesn't need his approval to get stronger.
At first, Naruto would probably be extremely shocked and in denial. He has always looked up to Kakashi as a mentor and friend, so the idea of being betrayed by him would be hard to accept immediately.
In some fanfictions, Kakashi might train Naruto by creating extremely challenging survival scenarios. For example, he could drop Naruto in a mock battlefield filled with various traps and enemies that Naruto has to overcome using his skills and wits. This forces Naruto to think on his feet and improve his combat abilities quickly.
Kakashi's reaction would be complex. He is a man who has lost many people in his life. But Naruto was special. He would hold Naruto tightly, his hands trembling slightly. He might try to use some of his own chakra to keep Naruto alive a little longer, but deep down he knows it's futile. His face would show a mixture of pain, regret, and love. He would remember all the times he trained Naruto, all the lessons he taught him, and how Naruto had surpassed all expectations.
Kakashi might immediately offer a comforting presence. He could put his hand on Naruto's shoulder and just be there silently, letting Naruto know he's not alone.