Nick is less sympathetic because the book does a much better job of selling Amy as the Cool Girl. In the movie, we never really see her as the cool girl and, in fact, in one of the first scenes we see an actual cool girl, Go, talking shit about her to Nick. When Amy eventually has the big "Cool Girl" monologue it's like. "When did you do any of those things?"

But in no way, shape, or form is it a Battle of the Sexes. To have a Battle of the Sexes, you have to have someone you might hold up as a representative of your gender - Nick and Amy are pretty much the textbook shitty versions of husband and wives (Amy obviously being more over the top.)