[Ajax search] Search custom post type

cpt.html

<form role="search" method="post" class="search-form padding-4" action="<?php echo home_url( '/' ); ?>">
	<label>
		<span class="screen-reader-text"><?php echo _x( 'Search for:', 'label' ) ?></span>
		<input type="search" class="search-field" placeholder="<?php echo esc_attr_x( 'Search...', 'placeholder' ) ?>" value="<?php echo get_search_query() ?>" name="s" title="<?php echo esc_attr_x( 'Search for:', 'label' ) ?>" />
		<input class="post_type" type="hidden" name="post_type" value="frequent" />
	</label>
	<input type="submit" class="search-submit button brand" id="searchsubmit" value="<?php echo esc_attr_x( 'Search', 'submit button' ) ?>" />
</form>

<script type="text/javascript">
	jQuery(document).ready(function($){
		// Bind the submit event for your form
		$('.search-form').submit(function( e ){ 
		    // Stop the form from submitting
		    e.preventDefault();
		    
		    // Get the search term
		    var term = $('.search-field').val();							    

		    // Get post type
		    var pt = $('.post_type').val();

		    // Make sure the user searched for something
		    if ( term ){
		        $.get( '/', { s: term, post_type: pt }, function( data ){
		            // Place the fetched results inside the #content element
		            $('#inner-content').html( $(data).find('#inner-content') );
		        });
		    }
		});
	});
</script>

functions.php

<?php

function template_chooser($template) {
	global $wp_query;   
	$post_type = get_query_var('post_type');   
	if( $wp_query->is_search && $post_type == 'frequent' ) { // Change post type here <--
		return locate_template('faq-search.php');  //  redirect to archive-search.php
	}   
	return $template;   
}
add_filter('template_include', 'template_chooser');