Hi John,
You would have to rebuild the header using GP’s Before Header Content hook and split it in 3 parts:
https://docs.generatepress.com/article/hooks-overview/
Try adding something like this:
<div class="grid-33 header-left">
<div class="site-logo">
<a href="URL TO YOUR WEBSITE" title="YOUR SITE NAME" rel="home">
<img class="header-image" src="URL TO YOUR IMAGE" alt="YOUR SITE NAME" title="YOUR SITE NAME">
</a>
</div>
</div>
<div class="grid-33 header-center">
Search widget shortcode here
</div>
<div class="grid-33 header-right">
Header right content
</div>
Then you can use a a plugin like this to turn search widget into a shortcode and add it in:
https://en-ca.wordpress.org/plugins/widget-shortcode/
Let me know if this helps.