logo centered in navbar with menu items floating to left and right.

The simplest way to achieve this is to add a custom link to your menu and replace it with your logo image. Put a slash (/) in URL box and “replace-me-with-logo” in “Link title” box. The menu will create something like this:

<li id="menu-item-519" class="menu-item menu-item-type-custom menu-item-object-custom current-menu-item current_page_item menu-item-519"><a title="Homepage" href="/">replace-me-with-logo</a></li>

Now all you need to do is run a preg_replace on the function that generates your menu and replace replace-me-with-logo with your img tag (your logo). The replacement function would look like this and needs to be added in functions.php of your child theme:

add_filter('tc_navbar_display', 'replace_custom_link_with_logo');
function replace_custom_link_with_logo($output) {
$html = '<img src="/path-to-your-logo-image" alt="your_company_name" />';
/* Ofcourse, you will need to replace "/path-to-logo-image" with the
 * actual path to your image file and "your_company_name" with your
 * company or website name... Make sure there is no unescaped single
 * quote anywhere inside the $html variable.
 */
return preg_replace('/replace-me-with-logo/', $html, $output);
}

Don’t forget to prevent the output of the normal logo:

add_filter('tc_logo_title_display', 'prevent_output_of_logo');
function prevent_output_of_logo($output) {
return;
}

Haven’t tested it, but it should work. If you have any trouble with it, get back here and I’ll make some tests and see what I missed. If it works, please mark it as resolved.