For the error: The BattlEye Service isn’t running properly: please restart your client
Solution taken from BattlEye FAQ:
https://www.battleye.com/support/faq/
“This is happening because you permanently declined BE installation by clicking “Do not ask me again.” in the BE Launcher window previously.”
Simply go to this folder (default location):
C:\SteamLibrary\steamapps\common\Conan Exiles\ConanSandbox\Binaries\Win64\BattlEye
Then run “Install_BattlEye.bat”.