I don't know what DAC you are using but mine has a Headphone Jack in addition to the RCA Phono sockets and this Headphone output is live when the RCA Phono output is live.
I have used a Bluetooth transmitter fed from the Headphone Jack to connect a Bluetooth Speaker. The Bluetooth Transmitter I used is powered from a USB Socket but there is a Gotcha! You cannot power the Bluetooth Transmitter from the RPi USB Sockets or from the same 5V Power Source as the RPi. If you do, you will get a very loud Hum on all outputs which I believe to be caused by an Earth Loop. Simply power the Transmitter from a separate USB Power Supply.
USB Transmitters are sold on eBay and Aliexpress for a couple of Pounds, including an Audio cable. for example :
https://www.ebay.co.uk/itm/194107588726?...1438.l2649
p.s. I should add that there is a considerable latency on the Bluetooth Speaker caused by the Bluetooth data buffering.
I have used a Bluetooth transmitter fed from the Headphone Jack to connect a Bluetooth Speaker. The Bluetooth Transmitter I used is powered from a USB Socket but there is a Gotcha! You cannot power the Bluetooth Transmitter from the RPi USB Sockets or from the same 5V Power Source as the RPi. If you do, you will get a very loud Hum on all outputs which I believe to be caused by an Earth Loop. Simply power the Transmitter from a separate USB Power Supply.
USB Transmitters are sold on eBay and Aliexpress for a couple of Pounds, including an Audio cable. for example :
https://www.ebay.co.uk/itm/194107588726?...1438.l2649
p.s. I should add that there is a considerable latency on the Bluetooth Speaker caused by the Bluetooth data buffering.
