"So if I used the record out ,I would just have to control the volume from the sub each time?"
Yes. You would have to use the volume on the Brio R to the speakers, then match the sub's volume to that. So you would be using (2) volume controls in that scenario.
Just run another set of speaker cables from the Brio R to the subwoofer, not from the speakers to the sub, but from the Rega to the sub.
Depending on the sub, if the sub has both speaker level inputs and outputs, you could run the Rega into the sub, then run your current bi-wire cables from the sub back out to the speakers.