Reply To: Disabling Live Search while Typing also disables auto suggest phrase click.

Well, the only way is using a custom code. Try adding this code to the functions.php file in your theme/child theme directory – make sure to have a full server back-up first for safety. For more details you can check the safe coding guidelines.

add_action('wp_footer', 'wp_footer_asp_trigger_on_suggest');
function wp_footer_asp_trigger_on_suggest() {
	?>
	<script>
	jQuery(function($) {
		$('body div').on('click', '.asp_keyword', function(){
			console.log('on');
			ASP.api(
				$(this).closest('.asp_r').data('id'), 
				$(this).closest('.asp_r').data('instance'),
				'searchRedirect',
				$(this).text()
			);
		});
	});
	</script>
	<?php
}

Unfortunately I can not guarantee this is going to work in all cases.