How to make child theme inherit parent custom theme options

It's extremely possible this question has been answered before, but I'm not 100% sure of my terminology.

The parent theme in question is a premium theme. One of the things it does is offer custom sections under Themes > Customise. I include an image so y'all know exactly what I'm talking about, in case I'm using the wrong words.

I need to overwrite some parts of the parent CSS, so of course I thought of a child theme. However when I load up my child theme, this menu area reverts to the default WP items.

As far as I can tell, the custom part of the parent theme is being loaded through the theme's functions.php, which means that it should be inherited by the child theme, right?

I've also tried to track down just inheriting the same settings as the parent theme, since for an interim solution that will work.

See this post here, which offers a solution to do just that. No joy there either though.

Does anyone have any ideas about what I should be pursuing to work this out?

EDIT: Child themes DO NOT inherit parents Customizer Options

Incidentally this is a site on WordPress.com, rather than self-hosted. First time I've dealt with WordPress.com so I'm not sure if that's significant or not.