I already had the same problematic and I found help from @bart who crafted a code for me. The code uses a CSS property that prevents the Body to scroll. It's when you put overflow:hidden to the body.
So basically it passes the body overflow:hidden when you click on something, and reverse it when you click on something else. It's maybe going to be an issue for you because of the many things you can click when the menu is open.
Add that script at page level in the second custom code zone.
Replace .project-item by the class of the object you click on to open the menu.
Replace .close-panel by the class of the oobject you click on to close the menu.
I'm going to ask if it's possible to put several classes at once.