Ahh the anchors are missing.
Try this instead:
add_action( 'generate_inside_mobile_header','lh_mobile_logos' );
function lh_mobile_logos() {
?>
<div class="site-logo mobile-header-logo">
<a href="https://SITE-URL" title="SITE TITLE" rel="home">
<img src="URL TO MOBILE HEADER LOGO" alt="" />
</a>
</div>
<div class="site-logo mobile-header-logo sticky-only">
<a href="https://SITE-URL" title="SITE TITLE" rel="home">
<img src="URL TO STICKY MOBILE HEADER LOGO" alt="" />
</a>
</div>
<?php
}