I have just successfully done what you want to do. I have no issues with my computer BT staying connected to Echo if I don't want it to, or reconnecting when I want it to. Here is the drill:
To begin with. my computer BT is in a state of being ready, but has been inactive for weeks (not connected to anything, incl. Echo).
So, first I needed to connect PC BT to Echo, so I said to Alexa, "turn on bluetooth". It replied "ready to pair, look in your BT device for signal EchoXXX". I click on Connect (It had been previously paired, but not connected). Alexa immediately bongs and said it was now paired to (device name). (Actually, the correct term Alexa should use here should be "connected" not "paired")
Next, I played music from a NAS on my network via my PC, and the sound came from Echo. Next step, is to disconnect it (not un-pair it.)
Without getting up from PC, I then clicked on Disconnect in the BT control and music stopped. Alexa bonged again and said "Disconnecting BT from (device name)".
I just went full circle, and my PC will not be connected to Alexa BT again until I do the above steps again.
OK, now try it yourself.