Well, it could strain their relationship. Avoidance usually creates distance, and in this case, Kakashi might feel hurt or confused as to why Naruto is avoiding him.
Their relationship is like a mentor - mentee bond. Kakashi has trained Naruto, and when he sees Naruto achieve great things, it can be overwhelming. For example, when Naruto masters a new jutsu that Kakashi thought was difficult for him, Kakashi might cry out of joy for Naruto's accomplishment.
It could become strained. Naruto snapping at Kakashi would create an initial rift between them. Kakashi might be taken aback by Naruto's outburst and Naruto, after snapping, might feel a bit guilty but also stubborn about his feelings, which would make the relationship tense for a while.
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.
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.