With help from Kenneth over at the Graphene support forum, I have already already done the following:
First, you'll want to hide the built-in menu with this CSS:
Second, install a PHP widget. (There are a few to choose from; just check the plugin repository.)
Third, create an action widget area in header.php in graphene_before_content
Fourth, add a PHP widget to that widget area, and copy the mix of HTML/PHP into it.
Fifth, add the CSS to your child theme stylesheet.
The problem is that you've already hidden anything with id="nav", so you'll want to change nav in your code to something else like mynav
Sorry to be a bother. Doing what I am able on my own and happy to repay the favor if I may.