Hack to hide TeamViewer dock icon, issue with restarting app
I solved the problem by running the following command in Terminal after killing the previous TeamViewer instance:

sudo launchctl load /Library/LaunchDaemons/com.teamviewer.teamviewer_service.plist


Next, I launch TeamViewer as usual. It shows up in the menu bar, as expected.

For those unfamiliar or uncomfortable with Terminal, the following AppleScript could be used, assuming TeamViewer is installed in /Applications:

do shell script "launchctl load /Library/LaunchDaemons/com.teamviewer.teamviewer_service.plist" with administrator privileges
do shell script "open /Applications/TeamViewer.app"