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After I disabled all of my other plugins, I then changed the theme to Twenty Sixteen and the menu now looks appropriate.

So it’s obviously an issue with how my theme is communicating with the plugin.

Yes, the same mechanisms that CMW uses to provide its functionality can be used by other plugins and themes as well. And they can get in each others way!
Is your theme a commercial one, or a free one? If it’s free, I can download it and see if I can find what’s occurring. Either way, it might be a good idea to put the question to the theme provider?
PS : It might be a good idea to re-enable your other plugins first, just to check that is really the theme and not one of those other plugins.

With the theme enabled and all plugins disabled, it was indeed an issue with the theme.

Since it’s a commercial theme, I just contacted you directly through your website to provide details for you to look into it, if you are able to.

Thank you!

Right…

The theme hooks into WordPress’s wp_nav_menu_args filter and, unless the menu being produced has a menu_id of “mobile-menu”, changes the “walker” to its own.

WordPress has a Walker class that is a general class for producing lists (which is all a menu is). WordPress also has a Walk_Nav_Menu class, which extends the Walker class specifically for menus. Usually plugins will provide their own “walker” as an extension of the WordPress Walker_Nav_Menu class. This is exactly what CMW does : it uses its own extension of the Walker_Nav_Menu class to do the filtering and set extra classes.
However, because your theme is intercepting every production of a menu and changing the requested “walker” class to its own, the CMW one never gets called. You’d probably find that other menu plugins (that have their own “walker”) would have exactly the same problem.

The theme code responsible can be found in tower/includes/core/codeless_megamenu.php, as part of the codeless_custom_menu class. The class is included into, and instantiated by, the theme’s functions.php as part of the theme setup. With the sole exception of the mobile menu that activates from the burger button (when the screen is small enough), every menu produced using WordPress’s wp_nav_menu() function will use the theme’s “walker”. (And even the exception isn’t really an exception : it’s just configured to use another of the theme’s “walkers”.)

I can guess what your next question will be : Is there a way around it?

By modifying CMW : No.
By modifying your theme : Yes. Add a check for $args[‘_custom_menu_wizard’] before deciding to switch over to the theme’s “walker”. However, this would be a change to the theme’s core code, and it would be lost (or would have to be re-applied) whenever the theme got updated! I would not advise changing any theme’s core code. If you have a child theme then that might be a different matter, but it would still mean that any theme update would have to be checked against your child theme modifications to ensure compatibility.

My suggestion : try dumping it in the theme provider’s lap?

I hope this helps.

Thank you for the very detailed response!

I have contacted the theme developer with your recommendation.

You’re welcome. And thank you!