Rename theme folder for MultiSite without losing customizations?

I have a MultiSite install, and am using a custom theme that I have created. I have named it ‘companytheme’. However, I would now like to split this theme up, and have ‘companytheme’ be the Basic version of the theme, and create a ‘companytheme-advanced’ child theme which will have more advanced functionality built on top of the Basic theme.

Is there any way to rename companytheme to companytheme-advanced and NOT lose my customizations done in the Customizer, Widgets, etc.? Having to re-set all of that back up would be time-consuming and annoying. I wish I’d have known this would be the path we’d take ahead of time and I’d have set two themes up to begin with. Keep in mind this is a MultiSite install, and all of the sites will share these themes. Some will be Basic, others Advanced. I don’t want to lose the customizations we’ve already made on any of them through the Customizer, Menus, Widgets… Hopefully there’s a way to change it in the DB?

Not sure if, to keep these across theme changes, it’s better to use options instead of get_theme_mod? New to using the Customizer.