Hi Chad, if a page header has content then yes it will replace the featured image.
Considering the search bar is common to all pages, you could place that content in the after Header Hook.
We can help with the markup and CSS if you want to go this route.
Hi Chad, you can add the code to the Dashboard > Appearance > GP Hooks – probably the After Header Hook is the best place to start. If you need a hand with a PHP conditional or some CSS then let us know. Might be worth using the GP .grid-container class for your content wrapper to keep it within the confines of the page content.