Good news is that when you have this set up by professionals, they will have a mixer that will allow a record out for the video guy to use :-).
Good (larger) PA’s, mostly of the digital kind, now allow you to create lots of outputs. While normally used for monitor use, they CAN be configured as stereo master outputs as well. This is handy if the music needs to be the same in all rooms but with varying and adjustable volumes. This can all be handled from one central place.
If different music is wanted in different rooms the story becomes a bit more complex, but nothing a pro-PA rental company can’t handle.