Perhaps James has more resources to take care of Harry. He could have a better living situation, more financial stability, or a stronger support system. In fanfiction, this can be a way to create a new dynamic for the characters, with James being the sole provider and protector for Harry.
This fanfiction scenario might also change the power dynamic. Instead of Harry being the 'lone hero' figure as in the original, he might be more of a 'son following in his father's footsteps'. James would be in a position of authority and protection, which would shape their relationship in a way that is quite different from the story we know, where Harry is more independent from the start due to the lack of parental figures.
One major challenge could be proving that they are the better parent for providing a stable and nurturing environment. Another might be dealing with the opposition from the other parent who also wants custody.
Maybe Sirius gets help from some powerful wizards or witches who believe in his innocence. They might testify on his behalf in front of the wizarding authorities. Sirius also could show his strong emotional bond with Harry. He has always been there for Harry in spirit, and now he can be there physically to raise him, which convinces the authorities to give him custody.
The plot could develop with Voldemort trying to use Harry to open the Chamber of Secrets again, believing that there are secrets within that could enhance his power further. Harry, on the other hand, might find allies among the Death Eaters who are not fully loyal to Voldemort and use them to create a revolt from within. Also, Dumbledore and the Order of the Phoenix would surely be trying to rescue Harry, which could lead to many intense battles and rescue attempts.
If Voldemort gets custody of Harry in fanfiction, it would be a huge deviation from the original story. Harry would probably be exposed to a lot of dark and cruel things. Voldemort could try to break Harry's spirit and make him abandon his values. There might be a struggle within Harry between the good he has been taught and the evil that Voldemort represents. And the wizarding world would be in even more chaos as Voldemort would have more control over the one person who was seen as the hope to defeat him.
Perhaps they are blackmailed. A powerful dark wizard could threaten to harm all of their friends and family if they don't abandon Harry. They may feel they have no choice but to sacrifice their relationship with their son to protect others they care about.
It might be related to power dynamics. If in the fanfiction, James has a lot of power and misuses it in a way that affects Harry or those around him, Harry would hate him. For example, if James uses his influence to push through unfair rules or to marginalize certain groups that Harry supports. Also, if the fanfiction changes James's personality to be more self - centered and less empathetic, Harry would likely dislike him.
Sirius has to figure out how to give Harry a normal childhood while also training him in magic. He can't just let Harry be unprepared for the wizarding world, but he also doesn't want to push him too hard. Also, he may face financial difficulties. Raising a child isn't cheap, and he might have to find ways to earn money while also taking care of Harry.
In a 'Harry Potter in a Custody Battle' fanfiction, it might be that Harry's muggle relatives and the wizarding community are in a custody battle. The Dursleys could claim they've been taking care of Harry all these years and have the right to keep him. But the wizarding world, like Hogwarts professors, might argue that Harry belongs with his own kind where he can learn magic and be safe from the threats only the wizarding world understands. There could be courtrooms filled with magic spells and arguments about what's truly best for Harry.
One possible reason could be a dark magic influence on James Potter in the fanfiction. Maybe he's been cursed or possessed, causing him to act out of character and disown Harry.