A issue that occurred has been fixed with some help from other plugin developers. This problem was that WordPress doesn’t have a hook to add fields to the Appearance > Menus page. I used to fix this by just overriding the function that created this page, but this way of working means more risk of conflicts with WordPress (updates / changes to the markup) and other plugins (only one can ultimately create the HTML that way). Thanks to the plugin author of Max Mega Menu, who pointed out a workaround other plugin developers have been using, this is no longer such an issue. I intended to implement their suggestion for some time now, and today I got around to it.
In the next release, my plugin now supports the wp_nav_menu_item_custom_fields hook and should be compatible with other plugins. See original support question here: Is there any way to use this with Max Mega Menus?
Those who need this functionality right away can grab the latest development version at the plugin page on WordPress.org