There could be a group of people who are against the idea of the feudal era and the magic associated with it interfering with modern times. They view Shippo as an unwanted intruder and Kagome as a traitor for bringing him into their world. This group would use all kinds of means, from political to technological, to try and get rid of Shippo and stop Kagome from continuing her adventures in her own time.
Maybe a demon that has somehow managed to cross over into Kagome's time without being fully noticed at first. This demon could be after Shippo for some reason, perhaps because of his connection to Inuyasha or his own unique fox abilities. As Shippo and Kagome start to figure out what's going on, the demon becomes the main source of conflict in the story.
Perhaps in the fanfiction, Shippo discovers a special power within himself that is related to Kagome's time. This power could be the key to helping Kagome solve a big mystery. Maybe there's an ancient artifact in her time that has a connection to the feudal era, and Shippo's newfound power allows them to understand and use it to their advantage. He could also make new friends in Kagome's time and learn valuable lessons about friendship outside of the feudal era group.
A time - traveling rogue from their own era who has followed them to Kagome's time. This character has his own agenda which involves causing chaos in different time periods. He sees Sango and Miroku as obstacles to his plans and tries to eliminate them. He might have some knowledge of modern technology from previous travels and uses it against them.
It could be a powerful dark spirit from Kagome's world that somehow manages to cross over to the Naruto universe and starts causing chaos. This spirit might be immune to normal ninja attacks and require Kagome's spiritual powers to be defeated.
It could be a demon who has a grudge against Kagome's father. Maybe her father defeated this demon long ago, and now that Kagome is in the picture, the demon sees it as a chance to get revenge on the family.
One possible plot could be that their own group betrays them. Maybe for some powerful magical item or a secret that Kagome holds. Shippo, being young and not as powerful, is easily overpowered. Kagome, on the other hand, tries to fight back but realizes she's been tricked by those she trusted. She then has to find a way to escape with Shippo and seek revenge or justice.
It could be Naraku. Even though the focus is on Kagome and child Bankotsu, Naraku is always lurking in the shadows in the Inuyasha universe. He might see Kagome's interaction with Bankotsu as a threat to his plans and try to manipulate the situation to his advantage. He could try to turn Bankotsu against Kagome or use Kagome's distraction with Bankotsu to carry out some other evil deeds.
It could be a jealous member of their group. Maybe someone who is envious of Kagome's special powers or the attention Shippo gets. This person might spread false rumors about them, leading to the rest of the group turning against them.
Inuyasha could be one of the main characters. Since he has a strong connection with Kagome in the original story, he might also be involved in the time - traveling adventure. Maybe he follows Kagome back in time or is already in the past waiting for her.
If the fanfiction is set in Columbia from Bioshock Infinite, Comstock would likely be the main antagonist. His fanatical religious beliefs and his plans for Columbia would clash with Kagome. Comstock wants to maintain his power and carry out his dark plans, and Kagome's presence, with her different way of thinking and her powers, would disrupt his carefully laid - out schemes.
One possible fanfiction could be about Kagome somehow traveling to the Naruto universe. Maybe she meets Minato and they form an alliance. And she discovers that Naruto is Minato's son. Shippo could be a unique addition, perhaps he has some special abilities that are useful in the Naruto world.