Instead of using the Elementor Header Template, you can do this instead:
Save your Elementor Header as a normal template.
Get the shortcode for that template.
Go to Dashboard > Appearance > GeneratePress > Make Sure Elements Module is Active.
Go to Appearance > Elements > New > Hook
In the hook content: add your elementor shortcode.
Hook: Select Header
Disable Site Header: Check
Enable Shortcodes: Check
Then go to display Rules and set where you want the header shortcode to be displayed.
More info on the Hook Element
https://docs.generatepress.com/article/hooks-element-overview/