You should split the writer share to whatever you agreed upon since you wrote a share of the song (as the producer, If it's agreed upon). if you are the sole publisher, you get that whole percentage of the publisher share.
It doesn't matter if you register the beat unless you've previously licensed it out, under another or the same title, to parties to use as is, the instrumental (even in the instance where an exclusive agreement requires all other parties associated with the licensed material to quit using, or surrender the material (you still gotta get those royalties from the instrumental already being placed for a time period already)).
What you have are 2 separate tracks (an instrumental and a song). Getting the song placement won't generate revenue for the instrumental, and vise versa.
So, unless you have licensed out the beat to, let's say, film/ tv, where they don't record over your beat and use only instrumental... you don't need to register it with a PRO/ PRA, just the complete song.