Rewrite post type slug only for child theme

Make the slug translatable when you register the custom post type:

register_post_type( 
    'post_type_name', 
    array (
        'rewrite' => array (
            'slug' => _x( 'post_type_name', 'URL slug', 'your_text_domain' )
        )
    )
);

Then create a small plugin for the site where you want to change the slug:

add_filter( 'gettext_with_context', 'change_my_slug', 10, 4 );

function change_my_slug( $translation, $text, $context, $domain )
{
    if ( 'URL slug' !== $context or 'post_type_name' !== $text or 'your_text_domain' !== $domain )
        return $translation;

    return 'changed-slug';
}