> I know now this device is "classic bluetooth" but I hoped / guessed that modern bluetooth implementations would also support that...

Alas the ESP32 BLE library does not do that. And also a quick low level test didn't work.

> The problem(s) I have now for several hours though: How can I guarantee that data transfer is finished before closing the request?

It turns out that it was not the reply but the handling of the request that prompted Firefox to always show "connection broken": One needs to actually read the request to satisfy Firefox.

With that in mind serving a captured image works ok. With an HTTP-Refresh it even updates continually.