I'm on Amazon tech support right now. Just sent them a link to this article so they can see the exact issue (since they keep thinking that I loose my Bluetooth connection which is not true). Let's see what they have to say. My speaker is a Logitech UE Boom, but I have the exact same issue.
By the way, if you go into the Alexa app when this happens you will see that the Bluetooth is still active.
I found this:
Apparently you can disable the auto-off on the soundlink mini when connected to the auxiliary port by pressing and holding the Bluetooth & + buttons together.
Not sure if this will help you. But I just tried it on my UE Boom to see if that helps. My speaker flashed twice after I did this so it did something.
Amazon tech support called today. Got me disconnect thru the Alexa App, the bluetooth speaker then re-connect the speaker. He then said he would call back in 40 minutes. 40 minutes later, he called and sure enough the speaker was no longer being used by my Echo Dot. We re-connected thru the Alexa App and the guy downloaded all the logs for the engineers.
Hopefully they will figure out how to keep the speakers active.