OK Michael, this is all a bit tricky, I admit, but it can be done, and once you get the hang of it, you may wonder why you had trouble at first. I just finished connecting one of my Taps to my TaoTronics BT transmitter that is plugged into the headphone out jack of my old Classic iPod, and right now my iPod is playing via BT to the Tap. Cool. It's the same principle to play from your Sonos assuming it has a headphone out jack.
First, know that you do not use the Alexa app on a mobile device to do the actually connecting, although you do use it to prep things. Also, you need to know three things about using your BT transmitter; how to turn it off, how to pair it, and how to clear any pairings in it that can be causing a connection problem.
You also will need to clear the BT cache of your Tap, so let's start there. But, first let's COMPLETELY turn off the Tap (not to standby, but all the way off) This is done by holding in the Tap's power button in the rear for a few seconds until you hear the three shut down beeps, then release the button. Wait a few seconds and then use the same button to turn it back on with a single press in; it will take a while for it to completely reboot.
Now, let's clear the BT cache of your transmitter. I do this with mine from a completely off state and holding in the power button of it for 15 seconds until all LEDs flash. Once I do this, the unit goes back to off state. Leave yours in same off state.
Now, back to your Alexa app on mobile device, go into Settings, find your Tap, and enter its settings. Select Bluetooth. You will see both Pairing Mode and Clear buttons. Tap the Clear button and the Tap will confirm the clearing action.
OK. both devices, BT TX and Tap are all cleared, and it is time to connect the two. At this point stop using the Alexa app. Tell the Tap to "turn on Bluetooth" or say "pair Bluetooth," either works. and the Tap will tell you, "from your mobile device look for your Tap's signal Amazon Tap-XXX on your mobile device."
NO! -- the transmitter is NOT a mobile device, so what you do instead is immediately turn on your BT transmitter's pairing mode (mine requires holding in power button for six seconds until I see alternating LEDs flashing), they both flash slowly and when the pairing takes place, the Tap announces that pairing has been successful. If your transmitter is plugged into an audio source, you will begin to hear the audio on Tap (be sure volumes of all devices are up high enough). That it!
My BT TX can retain the pairing much of the time, but if it doesn't, I may need to re-pair devices next time. Each gadget is different, and why, if I have not had them connected for a while, I need to refer to the user manual for BT TX. Also, my BT TX manual says to turn on it's BT pairing first, but I have more success to turn on the Tap (or Dot's) pairing first and then turn on the BT TX pairing last. See, all a bit tricky, but it works once the bugs are out of the set up procedure.
If you still have problems, let me know. These are of necessity long instructions, so you might want to print them out. Also, when pairing the two devices they need to be close to one another. I did all the pairing at a dining table today.
Good luck with it.
More Tips: turn off BT of other devices, like phone and tablets, etc. while doing this, and if you are having trouble connecting the phone and tablets to the Tap, your clearing the Tap's BT cache will help them connect, too. Also "forget" the Tap in the phone and tablet BT settings. Working with BT requires some detective work sometimes!