I don’t think that you can use the RPi 4’s Bluetooth capability to broadcast the Volumio sound output because the standard Volumio release does nor support it. However, I believe that I have a solution which will work with a Hat which provided both RCA and Headphone outputs such ad the IQaudio DAC+ or DAC Pro or this cheapo from Aliexpress PCM5122 Hifi Dac Audio Geluidskaart Module I2S + Led Indicator Voor Raspberry Pi 3B/2B - AliExpress 7
My plan is to use the RCA outpput to feed the local room Amplifier and Speakers and feed the Headphones 3.5mm output to a Bluetooth Transmitter like this one https://www.aliexpress.com/item/4000484636880.html?spm=a2g0o.cart.0.0.214f3c00TltplW&mp=1
… powered by plugging it into one of the RPi USB sockets and using the provided 3,5mm to 3.5mm cable to feed the audio signal from the Headphone output on the DAC Hat to the Bluetooth transmitter.
I have the parts on order from Aliexpress and would be happy to share my experience with you here.
Robin