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

Hi,

In that case, the keyword should be only filled to the input field, but it will not trigger the search or anything. Does it not fill the search input field either?

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.