Are the themes from Breakfast of Champions postmodern? Why does the postmodern presentation of this book work so well?
Yes, several aspects of Vonnegut's Breakfast of Champions are postmodern.
For example, postmodern literature is much more self-referential than most earlier traditions of literature. That very much applies here. Vonnegut includes references to himself as the author of the book, and comments upon himself. He is not part of an objective reality, distinct from the book and prior to it. Instead, he is as much part of a creative reality as the other characters.
Both the style and the content of this novel are skeptical. Postmodern thinking and literature is deeply marked by skepticism. There is no such thing as an unquestioned master narrative in either the thought or the literature, and Vonnegut questions all stories and symbols of his culture.
Postmodern literature breaks the surface, commenting on itself as a created work of art. That's true both in the content of this novel and in the presentation. Look, for example, at the odd definitions of common words and the simple, even crude, drawings.
Thematically, the lack of free will and the way that people's actions are determined by biology are quite postmodern.