So I have somewhat the opposite and it used to work! I had a simple PC style speaker system setup on the wire, and a receiver that has bluetooth. Typical use cases would just to let the wired speakers do their thing, but every once and awhile I wanted to hear it through the good speakers on the receiver. So I turn it on and BAM worked.... at least it used to.
Amazon did an update and now it seems the wire overrides the bluetooth (instead of just automatically ignoring it when bluetooth is connected). This happened around the time the new visual echo arrived (forget the name).
Anyway now when I have the echo connect to the bluetooth receiver while wired, it buzzes, freaks out for a couple seconds then just disconnects the bluetooth. If i try again with the wire disconnected, all is fine.
Of course Amazon doesn't have a help article for this, nor do they have an actual contact.