It's not clear if with "Live Update", you mean a Minor Update, that doesn't require to restart the sim, or a Major Update, which requires to exit, download and install the Stand-Alone Addon Manager, and restart the sim.
If you got a Minor Update notice, it's NOT POSSIBLE that "Couatl wouldn't work" *just* because of that because, the Minor Update DOES NOT update the Couatl program. Unless, with "does not work", you mean you got an error message from Couatl about some kind of problem, which you haven't said, so it would be best if you say if you saw any kind of message.
If, instead, you had a Major Update notice, if you should have downloaded and installed the latest Stand-Alone Addon Manager, in this case, have you got any errors while installing ?
But in any case, you should be more clear about what do you mean with "Couatl does not work" ? Do you have error messages ? If yes, which ones ? Is not starting ? Is crashing *after* starting ?