Hello and happy new year (sorry this took so long)
As I’ve read, what you describe seems in fact quite possible indeed (if I’ve understood well).
If what you want is a small vertical menu then you don’t actually need a hamburger menu at all, you just have to add a vertical menu –from the Add Pannel– specially for the mobile breakpoint and style it your taste to be small.
Then, from the Inspector Pannel, fix its position and dock it to the top: fixed elements don’t belong to any section (if anything they belong to the page).
Hope it solves what you were looking for.