If I own exclusive rights to a beat I purchased and copyright the song, do I include the producer in the copyright?

There is no set answer for this.

It depends on you and that persons agreement.

Usually if you 100% own a beat and there is no contact agreement that you have to give the person credit then no you do not have to include them. A contract would be a legal agreement

Sorry I can't really help but I'm really curious.

What do you mean by “a beat”? To me it would just be something involving (probably electronic) drums. Do you mean a backing track?

How are you going to “copyright the song”. What Song? Have you written a song? If so you already own the copyright.

If you're using the word “producer” correctly then no they wouldn't be involved in composing the song. A producer is simply the person in charge of the recording process. If you mean the person who who composed, played/programmed and recorded the backing track then they would be co-writer of the song. If you buy the exclusive rights to something it means that no one but you can use it. It doesn't mean you own the copyright to it.

I don't know why the mixing and mastering engineers would be involved in any copyright of the song unless they were involved in composing it.

Like I say though, I don't know what you intend doing to “copyright” the song. Be aware that copyright to a song is NOT the same as copyright to a recording of a song.

A lot of people seem to muddle them up snd use the word “song” to mean “recording”. If you write song you own the copyright on the song. If I pay for you to record it in (for example) my studio, I own the copyright on that recording of your song.