Customize existing menu item

Given that you want to include a picture in the wp_nav_menu output, your best bet is to use the wp_nav_menu_items filter, as suggested in the comments. This will allow you to do a search and replace on the html.

add_filter ('wp_nav_menu_items','wpse229358_filter_menu');
function wpse229358_filter_menu ($items,$args) {
  $search=""; // html string of button
  $replace=""; // html string of picture+name
  if (yourcondition) {
    str_replace ($search,$replace,$items);
    }
  }