Q: Could Not Connect to "Living Room ATV4"

Another user had trouble because they were logged into a Microsoft Domain. Could that be your problem?

Do you login into a Windows environment using that user ID, maybe accessing a work system remotely? I'm only asking because you said the problem was limited to one account on your computer, this was the cause of the other user's similar problem.

Nonetheless, there's something special about that user account that makes it incompatible with mirroring. You say it "could not connect", so it does see the ATV 4, and you are able to select it. Then either you cannot open the path to ATV, or ATV actively rejects the connection. I'm not sure if this Use AirPlay to display video from your Mac on an HDTV - Apple Support has any help, but it wouldn't hurt to check.

Those numbers in parentheses mean that your Bonjour environment is confused in some way. Those ATVs have reported in with different IP numbers, so the Bonjour service increments the number so they each have a different "name". What is your network configuration, and what router are you using? Do you disconnect the other ATVs from power when not in use?

Bonjour only affects whether you can see devices offering services, which you clearly can. I don't know what causes one particular user account to have difficulty, although there are other users with similar odd AirPlay problems.

Looking back over your posts, you mention TeamViewer. Now there's a long list of screen hacking software, like AirParrot, that are known to conflict with AirPlay. TeamViewer may be another example. Could you try uninstalling it? Be sure it's a completely clean uninstaall, so it removes any configuration changes, etc. If you ran TeamViewer from this account, it could be the problem.