The initial release of macOS Ventura and macOS 12.6.1 seems to have a bug that can cause Bartender and other apps to not be given Accessibilty and Screen Recording permissions after restarting macOS, even though they have been granted by the user, this results in Bartender showing its window requesting the permissions. To fix this as a first step which fixes it for many users I would try removing Bartender from the permissions using the “-“ button, then try re-adding it. If that doesn’t help fix it, you can remove Bartenders permissions completely and re-add them to do this Quit Bartender then run the Below comand in terminal:
tccutil reset Accessibility com.surteesstudios.Bartender; tccutil reset ScreenCapture com.surteesstudios.Bartender; tccutil reset Accessibility com.surteesstudios.Bartender-setapp; tccutil reset ScreenCapture com.surteesstudios.Bartender-setapp
This should then stop the issue happening again.