If your TV has Bluetooth, surely it can connect directly to it, as TV's can connect to soundbars by bluetooth, but that is just my logic working, not based on experience.
Bluetooth is a two way protocol.
If, on the other hand, your TV isn't a smart TV, or doesn't have Bluetooth, I don't see how you could "teach" your TV to send sound via Bluetooth. Again, that is just my logic, not based on experience.
What sort of Bluetooth transmitter were you told to buy? Do you have a link? That may help us help you from its description.
Might I also ask, does your version of echo have Bluetooth?
