Ah why didn’t I think of that. I actually have a custom header.php on all of the sites, mainly to account for <noscript> navigation. Within the custom header.php is the <?php do_action( 'generate_before_header_content'); ?>, followed by this code:
<?php if ( !empty( $generate_settings['logo'] ) ) : ?>
<div class="site-logo">
<a href="<?php echo esc_url( home_url( '/' ) ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home"><img class="header-image" src="<?php echo $generate_settings['logo']; ?>" alt="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" /></a>
</div>
<?php endif; ?>