Kakashi would likely be shown as a loving and protective husband. He would still be the skilled ninja we know, but he would also have a soft side for his OC spouse. He might share his past traumas with her and let her in on his more vulnerable side.
He'd probably be a mix of angry and sad. Angry at whoever or whatever caused Naruto to be in this state, and sad because he cares about Naruto so much. He might clench his fists tightly, trying to control his emotions while he focuses on helping Naruto. He'd also feel a sense of guilt, wondering if he could have prevented this from happening.
Kakashi may become more empathetic. By constantly watching over Naruto, he gets to understand Naruto's struggles better, which could make him more in tune with his own emotions and those of others.
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.
His character could develop in a more assertive way. Instead of just tolerating Naruto's behavior, he might start to set stricter boundaries. This could make him a more respected figure in the Naruto universe within the fanfiction. For example, he could demand more respect from Naruto and others who have been taking his kindness for granted. This would change his image from a laid - back mentor to a more authoritative one.
Their relationship would likely become very close. Naruto would probably start to see Kakashi as a father figure. Kakashi, in turn, would become more protective of Naruto.
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 would become more nurturing. In the original, he was a teacher, but as an adoptive father, he'd have to be more affectionate. He might show his face more often to Naruto, as a sign of trust, and be more open about his emotions.