centered-logo.php

<?php

function nore_home_logo() {	
  $centered_logo = '';
  $handle  = 'nore-style';
	$custom_logo_id = get_theme_mod( 'custom_logo' );
	$logo_render = wp_get_attachment_image_src( $custom_logo_id , 'full' );

	$centered_logo .= 'header .main-navigation ul > li.my-site-logo a {background-image: url(' . esc_url( $logo_render[0] ) . ');}';
	
	wp_add_inline_style( $handle, $centered_logo );
}
add_action( 'wp_enqueue_scripts', 'nore_home_logo', 200 );

How to use....

#1: Add the above code into your child theme's functions.php
#2: Create a menu via Appearance > Menu to include the Home link - move this link in the middle of your menu (Important)
#3: Add the class my-site-logo to the Home link and save

Now visit the Customizer and upload your site logo and this will render in place of the Home link. Style it to your needs and you have yourself a centered logo your single menu!

Your CSS would be something like this - adjust for your needs but leave this text-indent: -9999em; as is (important)

.main-navigation ul > .my-site-logo a {
        background-position: center center;
        background-repeat: no-repeat;
        text-indent: -9999em;
        width: 150px;
	height: 150px
 }

Note: This is a theme specific but the idea is the same for all menus. For Elementor you'll need to inspect your custom header to get the right CSS elements to target!