Exclude specific data when cloning a site?

There’s no way to do this within the plugin itself, but I suggest you approach it differently.

Option 1, preferred

The master site should *not* have a Piwik-ID. It should be empty.

In this way, every time you clone a new site, the cloned site is responsible for setting its own Piwik-ID.

Option 2

If the above is not an option, you could set or clear this value after the new blog is created.

In functions.php:

add_action( 'wpmu_new_blog', 'my_new_blog_defaults', 11, 2);
function my_new_blog_defaults($blog_id, $version=1){

	global $wpdb;

	if ($blog_id != get_current_blog_id()){
		switch_to_blog( $blog_id );
		$restore_blog_when_done = true;
	}

	// Here you go... could set the second parameter to "" to just clear the option.
	update_option('wp-piwik_settings', "...YOUR OPTION CODE HERE...");

	// We're done!
	// Restores previous blog after the switch_to_blog call.
	if (isset($restore_blog_when_done)){
		restore_current_blog();
	}
}