Ok, found it myself. After the cart has been loaded, you need to execute dropdown_menu('.nav:not(.nav-stacked):not(.nav-tabs)' , '> li', 'below', 100);
It would be great, if TB could publish some more documentation about BurnEngine or refer to other docs, if BE used other tech stacks for modification. It is by far the best theme engine and page builder for OC and it would be great to extend this engine with custom mods!
Yes, probably more efficient to just update the menu content and the item counter, but that would require a custom modification of the minicart function, which might need modification in future versions.
That way it was easier and quicker and served our needs for having an updated cart for html-cached pages via Varnish.