We have found the issue with the Install Loop, very sorry its taken a few days.
Unfortunately you will not be able to update automatically while this is happening, as such you will need to download the latest version from here and replace your current version and restart OS X.
It may be that it will still not install after trying the new version
If so you should try quiting Bartender 2.
Then running the below commands in terminal, then try again.
sudo rm -rf /Library/PrivilegedHelperTools/com.surteesstudios.Bartender.BartenderInstallHelper; sudo killall com.surteesstudios.Bartender.BartenderInstallHelper