By default, the post navigation option in the customizer only works for Posts post type. The following PHP snippet can be used to enable it in your CPT as well:
add_action( 'generate_after_entry_content', 'tu_cpt_footer_meta' );
function tu_cpt_footer_meta() {
if ( is_singular( 'your-post-type' ) ) : ?>
<footer class="entry-meta">
<?php generate_entry_meta(); ?>
<?php generate_content_nav( 'nav-below' ); ?>
</footer><!-- .entry-meta -->
<?php endif;
}
Learn how to add PHP here.
Make sure to replace your-post-type
with the name of your CPT.