I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide range of very creative and sometimes out - of - the - ordinary ideas, but this one seems rather unusual. You might try searching on some large fanfiction websites to see if it exists.
The chances are probably quite low. This is a very specific and rather unorthodox concept that might not be very popular among fanfiction writers. Most fanfictions tend to focus on more common themes like character development, new adventures, or different relationships within the bounds of the original story's feel.
One consequence is a change in his physical strength. Kurama provided Naruto with a huge boost in power, so without it, he might not be as strong in terms of brute force. However, this could also force him to become more strategic in combat.
Well, Naruto might gain new and powerful jutsu. Kurama could teach him some unique chakra control techniques that no one else in the village knows. This would make Naruto even more formidable in battle.
Naruto would likely become more powerful. With the Yin Kurama, his combat abilities could increase significantly. He might also become more mature as he has to deal with the added responsibility of this new power. For example, he would need to be more cautious in using it so as not to harm others accidentally.
If Naruto gets rid of Kurama in fanfiction, one way it could change the story is that Naruto would lose a major source of his power. He'd have to rely more on his own natural skills and training. This might make him seem more vulnerable at first, but it could also lead to him developing new and unique abilities. For example, he could focus more on his Sage Mode training and become even more proficient in it without Kurama's chakra interfering or overshadowing his progress.
One possible reason could be if Naruto in the fanfiction is being unfaithful to her. In the original series, their relationship is built on trust and love. So, if the fanfic portrays Naruto in a way that goes against that, Hinata might get mad.
If Hinata gets mad in the fanfiction, it can create a lot of drama. It might lead to a rift between her and Naruto, which could change the whole dynamic of their relationship. For instance, they might stop talking for a while and Naruto has to figure out how to make it up to her.
In some fanfictions, Sakura may make decisions without considering Naruto's opinion. Naruto values his friends and their input, so if Sakura goes ahead and does something like endangering the team or making a big decision on her own that affects everyone, Naruto could get mad. Also, if Sakura were to insult Naruto's dream of becoming Hokage, that would definitely be a reason for his anger as that dream is very important to him.