Different Site Logo on Sidebar/Topbar

You could always create a child theme to accomplish this. Perhaps the simplest way would be this:

* Create an empty directory in wp-content/themes/ called private.
* Add a file to private/ called style.css.
* Add the following to style.css:

/*
Theme Name: Private
Template: flat
Version: 1.0
*/

@media screen and (max-width: 800px){
.site-title img { display: none; }
}

Of course, change the 800px to whatever width less than which your logo image will be hidden.