You can create a custom template to show your search results, and use the querystring param in the insert pages shortcode to pass the relevant details.
First, in your theme, create your custom search results template (this example is based off of the twentysixteen markup, so adjust to match your theme):
<?php
/*
Template Name: Inserted Search Page
*/
?>
<div class="wrap">
<div id="primary" class="content-area">
<main id="main" class="site-main" role="main">
<?php
$search = new WP_Query( $_GET );
while ( $search->have_posts() ) :
$search->the_post();
get_template_part( 'template-parts/content', 'search' );
endwhile;
?>
</main><!-- #main -->
</div><!-- #primary -->
</div><!-- .wrap -->
Next, when you want to insert search results, use this form for your shortcode:
[insert page='any-page-on-your-site' display='your-search-template.php' querystring='s=teapot&post_type=product&type_aws=true']
Note that the inserted page can be any valid page on your site; your custom search template above will ignore whatever that is and use the querystring values instead to populate its WP_Query.
