Hey there!
Nice site you got there 
You could add some js to click on the menu toggle when the user clicks on one of the anchor links.
Give each link on the fullscreen menu a custom attribute in order to attach an eventHandler (The value is unimportant):
Paste this code at the end of the body tag, like above or below your jquery code:
let fullscreenLinks = document.querySelectorAll('a[toggle-menu]')
fullscreenLinks.forEach((fullscreenLink) => {
fullscreenLink.addEventListener('click', () => {
document.getElementById('menu-toggle').click()
})
})
Hope this works, only tested it in my devtools console!
IE doesn’t support forEach on nodeList, if you need IE support, maybe use a classic loop instead