Thanks - I had a look at Squeezelite-esp32 and found this mention:
Note that streaming to a Bluetooth speaker is not the main purpose and remains experimental, so your mileage will vary. We will not work on improving or fixing that feature, please don’t open issues about that.
The alternative seems to be an ESP speaker (ie. with the speaker integrated to the ESP board), like https://raspiaudio.com/produit/esp-muse-luxe. But that’s not exactly the use case I’m after.
I’m hoping that the latest bluetooth proxy update from 2023.7 enables exploring this use case further!