I've done way too much research into this, and can state categorically that there are several non-phone bluetooth transmitters that will work with the model 3. However, using them is such a pain, that I gave up on them, and just stream stuff from my phone. BTW I highly recommend an app called Neutron - it can play just about any format file you throw at it, sounds great, has all the bells and whistles, and just basically works right.
The first thing I got working was a AGPTek Rocker V2 bluetooth audio player. It's a tiny ipod-like device that can play many file formats to an audio out or via bluetooth. It sounds great via bluetooth in the car, but the car doesn't remember it when you restart the car, so you have to reconnect it every time you get in the car.
What I really want is a way to connect my trusty old ipod (upgraded 5.5 with flash drive and rockbox and a really big battery) to the car, so I bought a bluetooth transmitter from amazon. A small taotronics one. It didn't work at all. So I bought another, newer taotronics one - no workee also (sort of - it would connect in receive mode, but not transmit, which is what you want). So at this point I was getting frustrated, and just bought like 5 different ones of different levels and prices, and proceeded to try them all. The one that worked was from 1Mii, their B03 model. And it only kind of worked - you had to connect with it in receive mode, every time you started the car (as the car wouldn't remember the connection), and then switch it to transmit, and then it would work. That's way too much work though - you would end up sitting and fiddling with the things for 3-5 minutes every time you started the car, which is a complete buzzkill in a cool high tech car like this.
BTW I would happily use the USB audio interface if it wan't COMPLETELY ****ED UP. Seriously there are 10-15 year old cars from the early days of USB audio that work better than this. Thankfully the tesla support team has told me that a major upgrade is in the works for it.