Hi Sophie,
The issue is that the theme uses a script to prevent some event propagation/bubbling (I’m not exactly sure which and where), and that prevents the input focus.
Usually I recommend a custom code where possible, but in this case the event listeners are added via native javascript, where it is nearly impossible to remove event listeners for a specific node, it would cause even more issues.
I would rather recommend placing the search shortcode outside the menu scope, or before/after the menu.