Hi,
I see that you are copying the flags into the hamburger menu with the following code in /core/views/b2a3de771f/scripts/scripts.js file: $('#languageSelectorContainer').append($('.languageSelector').clone());
That code does not copy the events attached to the flags.
I recommend you to use Wrapper selector CSS in GTranslate settings page instead. You need to remove the language switcher, the code which clones it and set Wrapper selector CSS to be: .languageSelector,#languageSelectorContainer, in that case it will appear in both places and click events will also be fine.
Thanks! 🙂