Cannot select Bluetooth device from Sound Output
I will add the answer. 

Looking through discussions.apple.com, as usual, yields bad advice: reset, reformat, disk cleanup. Essentially do a bunch of random nonsense until the problem sorts itself out.

Before you do anything overly-complex or even irreversible. I recommend you try what I call the monkey's gambit:


Unpair
Forget Device
Pair
Repeat steps 1, 2, 3 a lot


(Look through comments for specific instructions)

Even when it seems like it's doing nothing, keep doing it. It took me 4 or 5 tries. I was nearly convinced it was the hardware, but then all of a sudden I'm back to listening to mediocre music on a pretty good bluetooth speaker.

This might have been expedited if I rebooted my computer, but I really didn't feel like doing that.

TL;DR please do not reformat or reset your computer without running the monkey's gambit for a while; it might save you a more major headache.

To be fair, this worked for me, but it is entirely possible your system is in a state where this will not help it. I do feel compelled, however, to claim that if you can see the device connected in the bluetooth menulet then it is probably not a hardware problem.