Change header.php of a specific WordPress Multisite

From what I understand you’re using one theme across multiple blogs inside your network? If that’s the case, you could edit the header.php to add a bit of php logic into it to check on what blog your header is currently loaded.

To make it work, you’ll need to get the ID of a blog you want to apply the change to (check for it in your network admin on a screen that lists all the blogs).

Then edit the header.php and add something like this:

<?php 
// 6 in my example is the blog ID that I want to apply the change to
if ( 6 === get_current_blog_id() ) {
    echo `<meta name="theme-color" content="#ff6600" />`;
}
?>

The other way would be to create a new child theme with a header.php file that has the additional meta tag and activate it only on a particular blog.

Hope that helps!