How do I get rid of the mirroring option icon when I don't have an apple TV (but continue seeing other mirroring options when available)?
If you don't want to connect to AppleTVs at all, you can disable the service that is constantly looking for AppleTVs:


Disable macOS' System Integrity Protection (SIP):


Reboot your Mac and hold Command-R as soon as possible. Let go when you see the Apple logo. In the menu bar at the top go to Utilities > Terminal

Now we will disable macOS' SIP, disable AirPlay and re-enable SIP again so your Mac stays safe. Enter these commands

csrutil disable

sudo chmod 000 /System/Library/CoreServices/AirPlayUIAgent.app/Contents/MacOS/AirPlayUIAgent

csrutil disable



Click the Apple logo in the upper left and select "Restart" 


Your Mac won't display the AirPlay Mirroring icon if an AppleTV is near you.

Let me know how this worked out for you.

edit:
To undo the changes use this command

sudo chmod 755 /System/Library/CoreServices/AirPlayUIAgent.app/Contents/MacOS/AirPlayUIAgent