Nintendo will have taken its time, but the latest system update (v13.0.0) finally brings Bluetooth audio compatibility to Switch consoles (including Lite and Oled versions).
Bluetooth audio and 2 controllers, no more!
Concretely, you can connect a headset, headphones or even a Bluetooth speaker. Be careful though because a maximum of two controllers, not more, can then be connected to the machine if you use this function.
Bluetooth and other constraints…
Other small constraints to take into account: the inability to use the Bluetooth audio function from two wirelessly connected consoles in multiplayer mode, the connection of only one audio device at a time and the incompatibility with the wireless microphones (therefore no Bluetooth headset / microphone).