Can you not use the casting feature on BSN rather than screen mirroring?
I've always cast from my phone to a chromecast dongle in one of the TV's a HDMI ports. By doing this, the stream goes straight to the TV, so you can use your phone to come on here or do anything else on it without affecting the broadcast on the TV screen.
Looking at the tutorial on BSN, you should be able to cast direct to a smart TV even if you dont use a dongle. The list of available devices should show when you tap the 'cast' icon. For some reason though, only my chromecast shows when I tap the icon on BSN. On YouTube, I have the choice of my TV, my EE box or my chromecast when I tap to cast.
For those not familiar, Casting & screen mirroring are 2 completely different processes. Screen mirroring uses the phones battery & CPU and replicates the phones screen on the TV. Casting diverts the stream directly to the TV, meaning that the phone is not involved in the process other than being a remote control.