I know I'm late to answer on this, but I find that many old forums for some are new ideas/problems for others, so here is my input:
As silly as the solution may be, I didn't see this in the forum and this is what I did.
I put my Bluetooth receiver on a smart plug and made a routine in Alexa that turns on the switch when I say, "Alexa, turn on home audio". When the switch turns on, the Bluetooth receiver turns on, and the echo automatically connects (as we've observed). When I'm ready to have audio back on the Dot again, I tell Alexa to "turn off home audio". This turns off the smart switch that the BT RCVR is on, which causes the Dot to disconnect, which allows audio to come out of the Dot again. It's a bit of a work around, adds the cost of a smart plug (about $7), but once you have it set up, it works like a charm....hope this helps someone