Hi alex, the search box can indeed be templated. Unfortunately the layout template path is not exposed (I believe this is logged for fixing, I'll double check)
However, there is a workaround, which is the ViewMap. This feature allows you to globally map a control to a template.
Take a look at these posts, the first walks through using the viewmap, the second shows an example of mapping the searchbox. keep in mind this will globally map the template so all instances of your search box will use this template
Customizing Sitefinity Controls with the ViewMap
5 Things: Useful Examples of the Sitefinity ViewMap
hope this is helpful!