FWIW, my Samsung soundbar has to be set to auto power-on. This is called "Bluetooth Power On" and the default is off. Once set, if the soundbar is powered "off" and a Bluetooth connection is then initiated, the 'bar will power on fully in Bluetooth mode. There may well be something similar in your soundbar, so best to check the manual. i don't think it's a problem with Echo, just that the soundbar isn't able to listen as it's fully powered down?
edit - ahh, just realised I missed your comment "the soundbar goes into full on mode by itself". That's weird and does suggest it's already in some sort of auto power-on mode. Have you tried a bluetooth connection from another device to test, eg mobile phone?