Hi Brad,
To make sure that we are on the same pape, we do not support conflicts between our theme and unsupported third-party plugins, but meanwhile, we are going to do our best to help.
The Pro Headers and Footers do not use the WordPress native header/footer feature and it is a separate functionality. The way that the third-party plugin has the URL as a query string might contribute to the problem.
To make sure if that is the underlying cause, please kindly go to Pro > Theme Options > Header and temporarily select the Switch back to Original Headers option:

The option above will unassign the Pro headers and will use the WordPress original header for the theme. Kindly check if the header shows correctly in that case. I did not risk to do it myself as your website is live and I do not have permission to do the test.
If yes, then it shows that the plugin is not compatible with the Pro theme header/footer system and you will need to either avoid using the Pro header builder or accept having no headers for that specific page.
As a side note, it is possible to assign the pro header manually to a page via code. The way it works is to install the Child Theme and add a code snippet below to functions.php file:
add_filter('cs_match_header_assignment', 'custom_search_header');
function custom_search_header($match) {
if (is_search()) {
$match = 25440; // the post ID for your header
}
return $match;
}
The 25440 is the ID of the header that you have set globally and I found it by going to edit the screen of the header in question and checking the URL of the browser address bar.
There is a conditional tag is_search() which is used for the search pages and is an example. You need to contact the third-party plugin developer and ask them to give you a piece of code that you can replace with is_search() to force the header to be assigned for that specific page.
I am not sure if it will work 100% and we can not implement the functionality for you, but I thought it will be a good starting point to open up a conversation with the plugin developer to handle the case.
Thank you.