Hello Hans

Its not working because whenever you update the cart it will not trigger any delivery cost method right now .

You need to add the code in website_sale module .

The code is below . either you can change the code and you can make a new module for inherit the website_sale mathod.

@http.route(['/shop/cart/update_json'], type='json', auth="public", methods=['POST'], website=True)
    def cart_update_json(self, product_id, line_id, add_qty=None, set_qty=None, display=True):
        cr, uid, context = request.cr, request.uid, request.context
        order = request.website.sale_get_order(force_create=1)
        order._check_carrier_quotation(order, force_carrier_id=order.carrier_id)
        value = order._cart_update(product_id=product_id, line_id=line_id, add_qty=add_qty, set_qty=set_qty)
        if not display:
            return None
        value['cart_quantity'] = order.cart_quantity
        value['website_sale.total'] = request.website._render("website_sale.total", {
                'website_sale_order': request.website.sale_get_order()
            })
        return value

Hope that this answer will help you.

Please don't hesitate to ask any question regarding this.

If you get the answer as you need then please accept the answer.

Thank You