I ran into this exact problem with these two components last week… it’s a little tricky because I intended the full-width component to be a temporary experiment, but we have nothing on the roadmap to integrate it by default so it’s sticking around longer than I expected.
The full-width component isn’t ideal because it requires changing the header template (the only way I could overcome some layout issues).
At a glance… I don’t think the template override in the header search component is necessary, so I can take a look at moving that to a widget decorator, which would avoid the problem.