Hi!
Thank you very much for the proper details, it helps me a lot 🙂 I managed to fix the issue.
The problem was, that the menu copies the search contents when it is clicked for the first time, causing all the event handlers to detach – meaning that plugin won’t work. It is probably designed for the built-in search box mainly.
Luckily there is an option on the plugin back-end, which tries to detect these changes, and then re-init the search function, I have enabled this for you here: https://i.imgur.com/4HtybZG.png
It should be working correctly now.