One way to develop an acquired taste for twilight fanfiction is to approach it as a form of creative exploration. Consider it as an expansion of the Twilight world rather than a replacement for the original. Start with short fanfictions to get a feel for how different authors play with the characters and settings. Try to find fanfictions that add new elements to the story, like new magical abilities or different species. As you read, think about how these new ideas fit into the existing Twilight framework. You can also take breaks between reading different fanfictions to let the ideas sink in. Over time, as you expose yourself to more and more of this unique form of storytelling, you'll likely develop an appreciation for it.
To develop an acquired taste for twilight fanfiction, start by choosing fanfictions that are somewhat similar to the parts of Twilight you already like. For example, if you liked the romance in Twilight, find fanfictions that focus on different romantic relationships within the Twilight universe. Also, don't be afraid to explore different writing styles. Some fanfictions might be written in a more experimental way. As you read more, you'll start to notice the nuances and grow to like it. Another thing is to read reviews or join fanfiction communities where people discuss these works. You can get a better understanding of what to expect and how to approach the reading.
First, give it a chance. Read a few chapters without any preconceived notions. You might be put off by some things initially, but keep going. Second, try to understand the author's perspective. Maybe they are trying to explore a different side of the Twilight world. Third, engage with other fans who like this type of fanfiction. They can give you insights that might make you see the value in it.