The overlay header style uses the search popup/overlay style by default for design/usability reasons, however, you can easily change that with a little code.
add_filter( 'wpex_menu_search_style', function( $style ) {
if ( wpex_has_overlay_header() ) {
$style = wpex_get_mod( 'menu_search_style', 'drop_down' );
}
return $style;
} );