Well, one possible plot could be that Hawkeye is initially a lone agent but gets recruited by Shield. He might be hesitant at first, but then discovers a conspiracy within Shield that he has to help expose using his sharp shooting skills and street smarts. Maybe he teams up with some classic Shield agents like Coulson and they go on a mission to take down a powerful enemy who is secretly infiltrating Shield.
One option could be a rogue agent from within Shield who has turned against the organization. This agent could be jealous of Hawkeye's newfound status in Shield and tries to undermine him at every turn while also planning some sort of major heist or attack on Shield's assets. They might have some sort of high - tech weaponry that Hawkeye has to outwit with his arrows.
On the other hand, if some of the fanfiction was a bit too outlandish or inappropriate, it could make things a bit awkward. Hawkeye might feel embarrassed, and the other Avengers might not know how to react. But they'd probably try to smooth things over and remind Hawkeye that it's just the creativity of the fans and not a true representation. They'd still support him as a teammate.
There are a few pirate novels with Hawkeye as the main character that I can recommend. Among them were One Piece: Demon Farm, One Piece: Supreme Swordsmanship, and New Pirate: Hawkeye Is My Defeat. These novels were all light novels or doujinshi novels with interesting plots. However, due to the limited search results provided, there might be other pirate novels with Hawkeye as the main character that were not mentioned.