A license agreement is exclusive if you are the only person who has the right to use the work as described in the agreement. For example, if you enter into an agreement with the owner of the song, Strangers in the Night for the exclusive use of the song in a movie, no one else could use the song in another movie. Exclusivity can be as narrow or as broad as you choose. For example, you could narrow the exclusivity by limiting the exclusive right to use Strangers in the Night in horror genre movies. Or you could expand the exclusivity of your license agreement by obtaining the exclusive right to use the song in movies and television shows.
Compulsory licenses are mandatory licenses established by law. A copyright owner must grant a compulsory license, provided that you follow certain legal rules and pay fees. For example, once Strangers in the Night has been recorded and distributed to the public on recordings, any person or group is entitled to record and distribute a cover of the song without obtaining the song owner's consent. Compulsory licenses are nonexclusive (meaning others can also cover Strangers in the Night, just like you can).
Long answer. So, the fact that a song has been exclusively licensed for a movie doesn't, by itself, preclude your ability to get the compulsory license for a cover version. But, keep in mind, if the song has never been "recorded and distributed to the public on recordings," you won't be able to obtain the compulsory license.
Example: A song was used in a movie but never released on a soundtrack or other recording format. It would not qualify for the compulsory license. You would have to wait until the song owner released the song on a recording distributed to the public.
To take advantage of the cover song compulsory license, a notice must be sent to the copyright owner along with a fee set by the U.S. Copyright Office, known as the statutory fee or statutory rate. The recording fee (in 2022) is 9.1 cents per song (or 1.75 cents per minute of playing time). Alternatively, you can use online services such as Songfile.