Two things come to mind… a plugin or theme conflict or a large number of entries on the page.
The easier of the two… how many entries are you listing on the page? If is a larger number, you can try limiting it to only 10 by using the
limit shortcode option. Here’s an example:
Set the limit and then try to Customizer. If it works then the resources allocated on the server for PHP were being exhausted when trying to run the Customizer of a page with lots of entries. Make your desired changes, save and publish them, and then remove the limit.
To determine if it is a theme or plugin conflict, here are links to the two relevant FAQs with more detail.
I hope this helps! Let me know what you find out.