hey man maybe in the customizer make the normal mobile menu show up
and as mobile menu style choose "desktop style" so it will just be a menupoint "cart"
you can also change the menu in appearance -> menus
i think the desktop style mobile menu style doesnt include any icons so it doesnt include the cart icon
if you want the cart icon just choose another menu style other than desktop menu style.
then use css to hide things except the cart icon