Menu with center logo and Shrinking Header

well first – take the logo center – navigation underneath.
this will end in : that navigation is in header_main_alternate container and not in logo container.
so this will be easy to set this (logo) to display: none (for the desktop screens)

On Dashboard – Appearance – Menu

the “navigation logo” is a custom link with img link in the navigation label ( something like : <img src="https://webers-testseite.de/weber/wp-content/uploads/2017/09/logo-top.png" alt="Logo" /> )
go and let show the classes for menu links – i gave a class to this logo: logoimg

so now the trick:
you have to edit : avia-snippet-sticky-header.js
on older enfolds this was part of avia.js but now we have an own external js file.

you can see it here (and download it too): https://pastebin.com/h4LW4JHV

on line 49. there is a new line:

navlogo         = $('.logoimg img'),

and this is in the script of shrinking the new usage of it on line 94:

navlogo.css({'maxHeight': 2*newH + 'px'});

so when we start at a normal header height of 100px – this navigation logo will start at 200px
and on normal shrinking amount of 1/2 this will end at 100px

to have a child-theme usage of this new file put this to your child-theme functions.php:

add_action( 'wp_enqueue_scripts', 'wp_change_sticky_header_script', 100 );
function wp_change_sticky_header_script() {
   wp_deregister_script( 'avia-sticky-header' );
   wp_enqueue_script( 'avia-sticky-header-child', get_stylesheet_directory_uri().'/js/avia-snippet-sticky-header.js', array('avia-default'), $vn, true);
}

put the file to child-theme/js folder

the rest now is quick css :

.logo, .logo a { display: none } 
.container.av-logo-container { position: absolute }
.logoimg .avia-menu-fx { display: none }

/*** this is the menu-item for the logoimg menu point - you have to look what item id it has ***/
#av-burger-menu-ul .menu-item-460 { display: none }

/*** some styling ***/
.logoimg img {
-webkit-filter: drop-shadow(2px 2px 5px rgba(0, 0, 0, 0.9));
filter: drop-shadow(2px 2px 5px rgba(0, 0, 0, 0.9))
}

@media only screen and (max-width: 767px){
.container.av-logo-container { position: relative }
.logo, .logo a { display: block }
}

now it depends on if you are having a sub-menu on top how to style the image: (you must know the menu-item id of the image)

#menu-item-460 img {
    top: -35px !important;
    position: relative;
}

but you got a good solution too now!

you only have to find a solution for smaller screens and for mobile view

Sorry i did not see that you are still working with Enfold 4.2.6. the avia-snippet-sticky-header.js. is part of Enfold 4.4 and higher.
Since the alb elements and a lot of js files are separated files. ( Due to optimization and only loading things you realy need for your setting)

Former times the shrink was done in avia.js . and so the old function: avia_header_size(). is still active .

so maybe an update could help.