Display search icon on mobile in burger flyout

Go and activate the shortcode for search in child-theme functions.php

add_shortcode('avia_search', 'get_search_form');

then you can place an custom link to your menu by Navigation Label: [avia_search]
maybe to select it easier give a custom class to that custom link.
The rest is css

but if you have that search on your main-navigation the magnifier symbol ( search ) is also on hamburger open visible !
Maybe you only had to style the colors of that Search icon

then you will not need anything else ? express your request a little more precisely.
__________
but if you want it like this: https://webers-web.info
you can hide that custom link in your “non-hamburger” navigation by setting it to display none.
Think of that custom class on that custom link:

#avia-menu li.menu-search {
    display: none;
}

now we got a menu item search – to have the chance to write into that input field – the link on the item itself disturbs – we had to get rid of the href :
this to child-theme functions.php:

function remove_href_from_search_input(){
?>
<script type="text/javascript">
(function($) {
	$('#header').on('click', '.av-main-nav-wrap', function() {
			$('#av-burger-menu-ul .menu-search > a:first-of-type').removeAttr("href");
	});	
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'remove_href_from_search_input');

on my testinstallation i got this in my quick css – but you had to adjust it to your settings and needs:

#avia-menu li.menu-search {
	display: none;
}

#av-burger-menu-ul li.menu-search a:hover {
	background-color: transparent !important;
}

#av-burger-menu-ul li.menu-search a:hover {
	opacity: 1 !important
}

#av-burger-menu-ul li.menu-search .ajax_search_response {
	margin-top: 30px !important;
}

#av-burger-menu-ul li.menu-search .ajax_search_response  * {
	color: #fff;
}

#av-burger-menu-ul li.menu-search .ajax_search_response .av_ajax_search_content .av_ajax_search_title {
	font-size: 16px
}

#av-burger-menu-ul li.menu-search .ajax_search_response  .av_ajax_search_entry:hover .av_ajax_search_title {
	letter-spacing:0.4px;
	color: #004e7f
}

#av-burger-menu-ul li.menu-search .av_ajax_search_image {
	background-color: #fff;
	border-radius: 0;
	color: #000 !important
}