The HDMI-splitter would need to route the bluetooth stream (probably an analogue stream once it leaves the Bluetooth receiver) to an HDMI-ARC connection. I’m not sure such a splitter exists.
I’d advise you to buy an Era 100 or Roam. Both can bring bluetooth streams into a Sonos set up. If you choose the Era you could even buy the Sonos USB-C adapter, so you’d have a cable connection to the speaker instead of a bluetooth connection.
Great thank you. Very helpful