Here's a fun fact: you can kill someone without LV raising.
In UTY, on the pacifist route, you fight Ceroba. At the end of her battle, her final request is that you take her life. You can spare her or kill her. If you kill her, your LV does not increase. I'm fairly certain it's because you didn't gain any Execution Points from it, it wasn't a thoughtless cold-blooded murder. She asked you to pull the trigger and end her suffering. Similarly, I don't think killing people in a war would raise your EXP or LV. Then again, it depends on the circumstances.