If Bartender 5 is not starting at login after updating from Bartender 4 it seems to be caused by macOS Sonoma being more restrictive with starting at login entries, and doesn’t like the name change from Bartender 4 to Bartender 5 (It keeps adding Bartender 4 to start at login as they share the same app ID even though its been deleted).
To fix this:
If this still doesn’t work, turn off Bartenders start at login, restart macOS and then try turning it on again.