I've created a sub-theme by copying the best_responsive folder and renaming it to mytheme_responsive, and replacing all instances of best_responsive with mytheme_responsive. In mytheme_responsive.info, I've listed best_responsive as the base theme.

It seems that when the theme-settings.php file exists in both theme folders, there are config options for both best_responsive & mytheme_responsive in Admin > Appearance > Settings. Changing settings for mytheme_responsive will not save & update, but it seems that changing the setting in best_responsive will affect both best_responsive & mytheme_responsive.

Ultimately, I ended up commenting out the theme_settings_alter function from best_responsive, removing the settings from best_responsive.info, and disabling the Best Responsive theme. Though it seems that editing the base theme should not be the solution. Should I not include best_responsive as the base theme?