Greeting Shanti,
I too have used the template tag menus by leveraging the extremely helpful Wagtail Demo site. In my case I didn't need a third level, but I did modify the menu in other ways. I was reticent at first, but I found that once I dug into the menu code, it really wasn't too hard to understand or modify. If you're comfortable doing so, I'd highly recommend rolling up your sleeves and deconstructing the Wagtail Demo example. If you run into any specific questions, I'm happy to help!
best,
-raymond