This guide will help you put your server on the Battlemetrics website as public.
With Password:
Go to Commandline Manager, and select Custom Commandlines.
Copy the settings image below and modify it to your preference.
Save and Restart the server to apply.
Without a password:
Make sure to follow the first step with the password as Battlemetrics requires you to have a "Public 1" setting for Valheim to show in their list.
Once registered, remove the password tick box and set it to "Public 0". Save and restart the server to apply.
To register your server in public in Battlemetrics, you must check their guide or steps to do so as it varies on their website if registration is required.