Replacing customizer.php in a child theme

The theme I am using (Radiate) has room for three Featured Pages. I edited the theme files to change this to four pages. I edited customizer.php as well to make sure live customization was still possible. No problems so far, worked like a charm.

My next step, however, was to restore the original theme files, and include the new options in my child theme. Therefore, I copied the original customizer.php into my child theme and made my adjustments to that file only. Unfortunately, it doesn’s seem to work that way (although a few other personalized php-files are working this way already). Is there a way to fix this, or is the only option to edit the original theme files?

Thanks in advance,

Joost Kerpels
http://joostkerpels.nl/rjso