Menu shrinking on scroll, but not ava_inside_main_menu

what kind of header-settings do you have: logo left menu right or something else?
what kind of content – an image as “second logo” ?

so – the script responsible for shrinking behavior is: avia-snippet-sticky-header.js
you can have a child-theme script instead of loading the parent one by:

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);
}

This script takes care of the shrink factor and other settings – for example the varialble line-height while shrinking.
In the function avia_header_size there is the calculation of the effect – the trick will be to include the new element to be calculated f.e. in height and width.

Was this advice enough to solve your problem or do you need support?
In this case I would have to know more about your setting – the best thing would be to see the page it’s about.
( Think of : i’m participant as you are – so i do not see private content in your postings )