First, it’s important to define job responsibilities in Hollywood. In television, the job of determining the music that accompanies every scene belongs to the showrunner. In features, it belongs to the director.
If it's a music cue for a song that everyone knows, that song's going to cost a fortune. The producer decides if we have enough money in the budget to license it. (By the way, when we license music, we often pay for what’s called “a needledrop.” If you’re young, ask your parents what that refers to.)
So, as screenwriter, you’re not doing your job. You’re doing someone else’s job, so that should answer that question.
But there are other reasons why you shouldn’t. Let’s think them through.