Kind of a cool feature – been thinking about how to do this lately.
Just as an idea, what about adding the logo as a menu item?
Go to “Appearance > Menus” and add a custom link.
In the Navigation Label, add the HTML for your logo:
<img src="URL TO YOUR LOGO" alt="" />
It will need some tweaking (removing on mobile etc..), but I wonder if that’s the right way to start.
Let me know 🙂