Another possibility is that there could be a rogue shinigami who doesn't like the fact that one of their kind is raising a human. This rogue shinigami could try to undermine Harry's growth and even try to harm him or the shinigami who is raising him.
The plot might revolve around Harry having to keep his relationship with the shinigami a secret from the wizarding world. As he grows up, he starts to notice how his upbringing by a shinigami gives him an edge in dealing with dark magic. He could use the knowledge of the shinigami to face Voldemort in a way that no one else could, like understanding the true nature of death and using it to his advantage.
The main antagonist could be a corrupted version of the Shinigami itself. The Shinigami that Naruto makes a deal with might be one of the good ones, but there could be an evil counterpart that doesn't want Naruto to succeed. This corrupted Shinigami would use all its powers and minions to stop Naruto from achieving what he set out to do with the deal.
Maybe it's a new villain created specifically for the fanfiction. This villain could have powers that are extremely dangerous to Naruto, like the ability to drain his chakra completely. So the Shinigami has to step in to save Naruto from this new and powerful foe.
A powerful rogue Shinigami could be the main antagonist. This Shinigami might be jealous of the Shinigami King's power and wants to overthrow him. He sees Naruto as a threat because of his connection to the king and tries to stop Naruto from fulfilling any tasks or missions related to the king. He could also be causing chaos in the living world by releasing dangerous spirits or curses.
The Akatsuki could also be the main antagonists. They are after Jinchuriki, and Naruto and Gaara being together would be a big target for them. They'd likely try all sorts of sneaky tactics to capture them, which would put Naruto and Gaara's friendship and skills to the test.
In this type of fanfiction, Voldemort could still be the main antagonist. But the way Harry approaches him would be very different. With Riddick's influence, Harry might see Voldemort as just another powerful enemy to be taken down in a more direct, perhaps violent way, rather than through the more magical and somewhat convoluted ways in the original series.
It could be an evil Gundam pilot who is jealous of Naruto's unique abilities because of the Shinigami mark. This pilot might see Naruto as a threat to his own power and status in the Gundam world and constantly tries to undermine him in battles.
Maybe it's another dragon, a malevolent one, who sees the female dragon's act of raising a human as an abomination. He decides to target Natsu to prove a point or to gain power over the female dragon. The malevolent dragon could try to undermine Natsu at every turn, creating numerous challenges for him.
A power - hungry merman who wants to rule over all the mer - people. He sees Harry as a threat because Harry has some special powers or a claim to the throne that he doesn't want to share.
A new dark wizard who has emerged from the shadows, someone who idolized Voldemort and is trying to pick up where he left off. This antagonist could be using new and more insidious magic that Harry has to figure out how to combat as an Auror.