checkout page refresh issue

I think we don't need to reload the total summary. Because, when click the Next button, Magento will make a request(API) V1/carts/mine/shipping-information to re-calculate the totals and output the totals data to our templates.

So, if we want to check the fee, we should check the response total_segments

When click Next to the payment step, there is a request to set shipping information vendor/magento/module-checkout/view/frontend/web/js/view/shipping.js

             /**
             * Set shipping information handler
             */
            setShippingInformation: function () {
                if (this.validateShippingInformation()) {
                    setShippingInformationAction().done(
                        function () {
                            stepNavigator.next();
                        }
                    );
                }
            }

This request will re-calculate the totals.

In your case, in our html template, it should have a isDisplayed() function:

Kensium/DeliverySign/view/frontend/web/template/checkout/cart/totals/fee.html

<!-- ko if: isDisplayed() -->
<tr class="totals fee excl" data-bind="visible: canVisibleDeliverySignBlock">
    <th class="mark" colspan="1" scope="row" data-bind="text: title"></th>
    <td class="amount">
        <span class="price" data-bind="text: getValue()"></span>
    </td>
</tr>
<!-- /ko -->

Check isDisplayed() function:

Kensium/DeliverySign/view/frontend/web/js/view/checkout/cart/totals/fee.js

define([
    'ko',
    'uiComponent',
    'Magento_Checkout/js/model/quote',
    'Magento_Catalog/js/price-utils',
    'Magento_Checkout/js/model/totals'

], function (ko, Component, quote, priceUtils, totals) {
    'use strict';
    var show_hide_deliverysign_blockConfig = window.checkoutConfig.show_hide_deliverysign_block;
    var delivery_sign_amount = window.checkoutConfig.delivery_sign_amount;

    return Component.extend({

        totals: quote.getTotals(),
        canVisibleDeliverySignBlock: show_hide_deliverysign_blockConfig,
        getFormattedPrice: ko.observable(priceUtils.formatPrice(delivery_sign_amount, quote.getPriceFormat())),

        isDisplayed: function () {
            return this.getValue() != 0;
        },
        getValue: function() {
            var price = 0;
            if (this.totals() && totals.getSegment('fee')) {
                price = totals.getSegment('fee').value;
            }
            return this.getFormattedPrice(price);
        }
    });
});

This function will check the totals fee segment from the response.

I make a git pull here.

NOTE: Make sure your fee is calculated right way. On the payment step, please check the response has our fee.