There would also be challenges in terms of the baby's protection. Both the Order of the Phoenix and the remaining Death Eaters could see the baby as a threat or a tool. The Order might fear that the baby could be turned to the dark side, while the Death Eaters might want to use the baby's power for their own evil purposes. Harry and Bellatrix would have to constantly be on guard to protect their child.
The challenges would be numerous. Socially, he would be an outcast among those who oppose the Dark Arts. His magical education would be limited to Bellatrix's knowledge, which is mainly centered around dark magic. This could put him at a disadvantage when dealing with more complex and positive forms of magic. Emotionally, he would lack the warmth and support that a normal family provides, and this could lead to serious psychological issues.
Financial challenges could be one. They are still young, and having a baby means they need to provide for it. Buying things like diapers, food, and clothes can be costly. Also, they might face challenges in balancing their own hobbies and dreams with taking care of the baby. Phineas loves his big building projects, and Isabella has her own interests, but now they have to make time for the baby too.
Maybe she wants to humiliate him. By treating him like a baby, she can make him seem weak and powerless.
One challenge would be Harry's strong will. He has his own beliefs and values that are opposed to Bellatrix's dark ideology. Another challenge could be hiding Harry from Voldemort. Voldemort might not approve of Bellatrix raising Harry in her own way. Also, Bellatrix herself has a reputation to uphold among the Death Eaters, and raising Harry could be seen as a risky and strange move.
One challenge could be the acceptance of the baby in the elven community. Some elves might be traditional and not fully accept a baby born from their relationship. Another challenge would be protecting the baby from the dangers of Middle - earth. There are always threats like orcs and other dark forces. Also, they would need to balance their duties as elves with raising their child.
I'm not sure if there is such a specific fanfiction. It's a rather unusual and probably not very common concept within the fanfiction world. The general fanfiction often focuses on more mainstream relationships and storylines in the Harry Potter universe.
One big challenge would be trust. Bellatrix has been a loyal Death Eater and has tried to kill Harry many times. Even in pet form, Harry would have a hard time believing she wouldn't turn on him if she regained her normal form. Another challenge would be how others in the wizarding world would react. The Order of the Phoenix might not trust Harry's decision to keep her around, even as a pet.
One challenge would be time management. They are both busy shinobi. Naruto has his Hokage duties and Sasuke is often away on missions. Finding time to take care of the baby would be tough.
It would be very strange and likely a very dark fanfiction. Bellatrix is a Death Eater and Hermione is a good - hearted Gryffindor. The idea of Bellatrix being involved in delivering Hermione's baby goes against their normal character dynamics. It might involve a very convoluted plot where some sort of dark magic or forced situation has led to this scenario.
It would be extremely strange. Harry is known for his connection to the light side, while Bellatrix is one of the darkest characters. If he were their son, perhaps he would struggle with his own identity. He might have been raised in a very different environment, surrounded by Death Eaters, but still have some innate goodness in him that would cause internal conflict.