It will not work they way you want it to. It is either/or
You need a setup like I have for my stereo, where I have a dedicated Dot to feed the stereo with music from Dot and the Echo is left alone as is. Dots are cheap so buy one today.
Someone may tell you how to place an audio switch in, but that is a hassle. My Dot on stereo is on one input switch and other devices are other switches, so it is easy to change between inputs on stereo switches.