Using an external speaker on iPad while connected to an external display

Hey guys! I recently bought an external monitor, and I'm in the process of figuring out, how all this works on an iPad (I have a non-M1 iPad Air, iPadOS 17.2).

The thing I noticed is that when you connect to an external display, you're kind of stuck with its built-in speakers. The only exception I found is when using the AirPods: here the iPad automatically figures out that I'm using AirPods and moves the audio to it.

But what if I have an external bluetooth speaker and want to use it? When I connect it to the iPad via Bluetooth, the audio channel is still "stuck" on the external display. The only alternative is to connect the speaker directly to the display (not to the iPad) using a cable, and then it all works. But that doesn't seem like a modern streamlined solution :)

So my question: is it possible to manually change my audio channel to this bluetooth speaker, while I'm connected to the external display?