Ghost Plugins strikes again: https://www.ghostplugins.com/secret/841l164f
I think you may need a business account to use this SS feature.
P/W for GhostPlugins secret plugins is "whisper"
I highly recommend subscribing to their list, they are awesome!
From your Squarespace account, go to Site Code Injection. Copy and paste the code below into the Header Code Injection box.
<script>
Y.on('domready', function(){
var mobileSocial = Y.one(".Header-social").cloneNode(true).addClass('.mobile-social');
Y.one(".Mobile-overlay-menu-main").append(mobileSocial);
});
</script>