I'm pretty sure I remember reading last year that you cannot do both Bluetooth tether to a speaker and multi room music simultaneously, that it is a technical limitation of Bluetooth, and it will never be otherwise. I'm pretty sure it was Amazon that said this. So you'll have to go wired, or nothing.
Edit: now that I think about it, the reason it can't work is because of the inherent LONG latency in Bluetooth.
30+ years of using Bluetooth and it's still crap.