How to sync one menu item into other menus?
My suggestion is to use hook_form_FORM_ID_alter to add a custom submit handler. In that submit handler, you will get the current menu item values. Use those values to create a new menu item for the other menu.

Functions to use:


hook_form_FORM_ID_alter. Here form id will be 'menu_edit_item'
Use form['#submit'][] = "custom submit handler" in form alter.
Use menu_link_save in custom submit handler based on the values you get form the form_state.