enqueue script during add menu items means before save menu items

Because this menu items are appended with ajax response you need to run this JS functions again for each item that you add to the menu with the document#menu-item-added event.

Add something like this in your JS file.

$(document).on('menu-item-added', function(event, markup) {
    // debug that the hook work
    console.log(markup);
    $.each(markup, function(index, menuItem) {
        if(menuItem.id) {
            // check menu item html markup
            console.log(menuItem);
            $('#' + menuItem.id).find('.hpemenu-fonticons').fontIconPicker();
            $('#' + menuItem.id).find('.fonticons-iconwpcolorpicker').wpColorPicker();
            $('#' + menuItem.id).find('.bsnselect').select2();
        }
    });
});

Add click event with function to run where you click in the document ready and in after menu item added.

So Your code should look like this:

jQuery.(document).ready(function($) {

    $('.hpemenu-fonticons').fontIconPicker();
    $('.fonticons-iconwpcolorpicker').wpColorPicker();
    $(".bsnselect").select2();

    // Add click event
    $('a.item-edit').click(some_function);

    $(document).on('menu-item-added', function(event, markup) {
        // debug that the hook work
        console.log(markup);
        $.each(markup, function(index, menuItem) {
            if(menuItem.id) {
                // check menu item html markup
                console.log(menuItem);
                $('#' + menuItem.id).find('.hpemenu-fonticons').fontIconPicker();
                $('#' + menuItem.id).find('.fonticons-iconwpcolorpicker').wpColorPicker();
                $('#' + menuItem.id).find('.bsnselect').select2();

                // Add click event
                $('#' + menuItem.id).find('a.item-edit').click(some_function);
            }
        });
    });

    function some_function() {
        var itemId = $(this).closest('li').attr('id'),
            menuItemType = $('#' + itemId + ' .field-menuitem-type input');

        var showHideMenuItem = function() {
            //Some codes here
        };
        showHideMenuItem();
        menuItemType.on('change', showHideMenuItem);
    }
});