It could be a rogue Amazon who disagrees with the idea of an outsider like Captain America being on Themyscira. She might see him as a threat to their way of life and try to drive him out, or even worse, harm him.
Maybe an old enemy of Captain America follows him through the rift to Themyscira. This enemy wants to use the resources or knowledge of the Amazons for his own evil purposes. He could try to manipulate some of the Amazons to turn against Captain America, causing chaos on the island.
The Night King could be a great antagonist. His army of the dead and his own powerful magic would pose a huge threat to Captain America. Cap's used to fighting human foes, but the Night King is something entirely different.
One idea could be Captain America arriving on Themyscira and being initially met with suspicion by the Amazons. But as they learn of his values of justice and courage, they start to respect him. He could then train with the Amazons, sharing his combat techniques while also learning from their unique fighting styles.
Maybe it's an old enemy who has somehow survived and is now more powerful. They could be seeking revenge on Captain America for past defeats. For example, Red Skull could have found a new way to return and cause trouble.
The main antagonist could be a powerful sorcerer. As I mentioned before, in the context of a magical curse causing the weight gain. This sorcerer could have his own motives, like trying to take over the world by first weakening the Avengers' most iconic figure. He could use his dark magic to mess with Captain America's body and make him gain weight, and then use this as a distraction while he executes the rest of his evil plan.
Maybe there's an Amazon who has turned against her sisters and sees Batman's arrival as an opportunity to gain power. She could try to use Batman's technology for her own evil plans, and Batman would be caught in the middle between her and the rest of the Amazons.
Maybe a group of bounty hunters hired by the remaining Imperial forces. They see Luke as a big threat and Captain Rex as an annoyance. They keep hunting them down throughout the story, creating all sorts of dangerous situations for our heroes.
He could also meet Odin. Odin might view Captain America as an interesting mortal from Midgard and might have some words of wisdom or a task for him to complete during his stay in Asgard.
An alien race with a grudge against the Republic could also be the main antagonist. They might target Rex specifically because of his strategic importance and his role in leading many successful missions against them in the past. His death at their hands would be a major blow to the Republic's morale.
Thanos could still be a great antagonist even in a fanfic. He could come back with a new plan to obtain the Infinity Stones again. His powerful and menacing presence would create a high - stakes situation for Captain Marvel and the Avengers to face.
A good antagonist could be a rogue S.H.I.E.L.D. agent who has his own plans for power. He might see Captasha and Black Widow as threats to his scheme. This agent could have access to some high - tech weapons and a group of loyal but misled followers.