Hi there,
you could add this using a Hook Element:
https://docs.generatepress.com/article/hooks-element-overview/
The after_header
hook will place it in same place. Then set the display rules for the entire site.
Markup something like this:
<div class="site-name-bar">
<div class="grid-container">
<!-- Your HTML -->
</div>
</div>
The inner DIV uses the GP grid-container
class to maintain the inner content width.
Then you can hit the custom site-name-bar
class for your background and padding.