Given that I do not have Premium members at this time for my client's site, how do I remove the Premium Membership link found at:

Member Module > My Settings > Member Settings.

If you open it up in the builder, you'll notice it's set as follows — Type: Misc, Format: Skin Element, Parameters: settings_submenu. You can use this to fetch any skin element from the templates straight into the builder.

This particular snippet is located in /pro/modules/skin_members.php, as it's unique to the Members module, and distinct from dynamic templates (which would go to the _ow file). Edit there as necessary.