About 13 minutes in, the main character states, "Apparently before Mr. Cerf was ever famous, he started Random House books which happens to be the company that put The Catcher in the Rye in print." Little Brown was, in fact, the publisher.
Although Jamie has grander ambitions for his play, all he really needs is permission to put it on one time as a prep school production. He never actually makes this clear to Salinger, who could conceivably not care about such a minor use of his novel; instead, the conversation the two have makes it seem like Jamie was asking for greater rights to the novel.