Change WooCommerce Checkout shipping method cost from a WC Session variable

There are multiple mistakes and missing things in your code…

I have commented/disabled the following code line, as you don’t provide distancecal.php file:

// include_once(plugin_dir_path(__FILE__) . 'distancecal.php'); // Include PHP functions file

Also, you should not use WooCommerce existing Ajax actions, so I changed your action name from update_checkoutto shipping_cost_update.

Then there was some missing mandatory hooked functions (see this thread)

Try the following revised main php code:

<?php
/**
 * Plugin Name: Woo Custom Shipping Value (Test)
 * Description: Customizations for WooCommerce.
 * Version: 1.0
 * Author: udd_ish
**/

defined( 'ABSPATH' ) or exit;

add_action('wp_enqueue_scripts', 'enqueue_custom_scripts');
function enqueue_custom_scripts() {
    // Enqueue the custom JS file
    wp_enqueue_script('jquery');
    wp_enqueue_script('distancecal-scripts', plugin_dir_url(__FILE__) . 'js/distancecal-scripts.js', array('jquery'), '1.0', true);
    wp_localize_script('distancecal-scripts', 'ajax_object', array('ajaxurl' => admin_url('admin-ajax.php')));
}

function modify_shipping_cost($shipping_cost) {
    if ( isset($shipping_cost) ){
        WC()->session->set('shipping_cost', $shipping_cost);
    }
    return $shipping_cost;
}

// Add a button after the billing form in WooCommerce checkout
add_action('woocommerce_after_checkout_billing_form', 'add_custom_button_after_billing_form');
function add_custom_button_after_billing_form() {
    // include_once(plugin_dir_path(__FILE__) . 'distancecal.php'); // Include PHP functions file
    ?>
    <div id="custom-button-wrapper">
        <button type="button" id="custom-button" class="button alt"><?php _e('Calculate Delivery Fee'); ?></button>
    </div>
<?php
}

add_filter( 'woocommerce_package_rates', 'shipping_package_rates_filter_callback', 100, 2 );
function shipping_package_rates_filter_callback( $rates, $package ) {
    // HERE define your targeted rate id
    $targeted_rate_id = 'flat_rate:4'; 
    
    if ( ! array_key_exists($targeted_rate_id, $rates) ) 
        return $rates;

    $new_cost = WC()->session->get('shipping_cost'); // Get new cost from WC sessions

    if ( ! empty($new_cost) ) {
        // (optional) Customize the shipping method label name when the new cost is set
        $rates[$targeted_rate_id]->label = __('New name', 'woocommerce'); // Set the new name

        $rate = $rates[$targeted_rate_id];
        $cost = $rate->cost; // Set the initial cost in a variable for taxes calculations
        $rates[$targeted_rate_id]->cost = $new_cost; // Set the new cost

        $has_taxes = false; // Initializing
        $taxes     = array(); // Initializing
        foreach ($rate->taxes as $key => $tax_cost){
            if( $tax_cost > 0 ) {
                $tax_rate    = $tax_cost / $cost; // Get the tax rate conversion
                $taxes[$key] = $new_cost * $tax_rate; // Set the new tax cost in taxes costs array
                $has_taxes   = true; // There are taxes (set it to true)
            }
        }
        if( $has_taxes ) {
            $rates[$targeted_rate_id]->taxes = $taxes; // Set taxes costs array
        }
    } else {
        unset($rates[$targeted_rate_id]);
    }
    return $rates;
}


// PHP WP AJAX handler
add_action('wp_ajax_shipping_cost_update', 'shipping_cost_update_callback');
add_action('wp_ajax_nopriv_shipping_cost_update', 'shipping_cost_update_callback');
function shipping_cost_update_callback() {
    $test_value = 100;

    wp_send_json( array( 'updated_shipping_cost' => modify_shipping_cost($test_value) ) );
    wp_die(); // Make sure to exit after handling the AJAX request
}

And for distancecal-scripts.js JavaScript file:

jQuery( function($) {
    // Add your custom JavaScript here
    $(document.body).on('click', '#custom-button', function() {
        const streetAddress = $('#billing_address_1').val();

        if ( $('#billing_address_1').val() !== '' ) {
            $.ajax({
                type: 'POST',
                url: ajax_object.ajaxurl,
                data: {
                    action: 'shipping_cost_update',
                    street_address: streetAddress
                },
                success: function(response) {
                    console.log(response);
                    $(document.body).trigger('update_checkout', { update_shipping_method: true });
                    
                },
                error: function(error) {
                    // Handle AJAX error if needed
                    console.error('AJAX error:', error);
                }
            });
        }
    });
});

PHP: Missing mandatory code (Refresh shipping methods cache):

// Refresh shipping methods
add_action('woocommerce_checkout_update_order_review', 'checkout_update_refresh_shipping_methods' );
function checkout_update_refresh_shipping_methods() {
    if ( WC()->session->__isset('shipping_cost') ) {
        $packages = WC()->cart->get_shipping_packages();
        foreach ($packages as $package_key => $package ) {
             WC()->session->set( 'shipping_for_package_' . $package_key, false );
        }
    }
}

// Remove the WC session variable once the order is created
add_action('woocommerce_checkout_order_created', 'remove_wc_session_shipping_cost_variable', 10, 1);
function remove_wc_session_shipping_cost_variable( ) {
    if ( WC()->session->__isset('shipping_cost') ) {
        WC()->session->__unset('shipping_cost');
    }
}

Add the code (both functions) to your main plugin file.

Optional code: Reset/remove the WC Session variable if customer goes other pages.

// (optional) Remove the WC Session variable on other pages than checkout
add_action('template_redirect', 'wc_session_shipping_cost_variable_only_on_checkout', 10, 1);
function wc_session_shipping_cost_variable_only_on_checkout() {
    if ( ! ( is_checkout() && ! is_wc_endpoint_url() ) ) {
        remove_wc_session_shipping_cost_variable();

        $packages = WC()->cart->get_shipping_packages();
        foreach ($packages as $package_key => $package ) {
             WC()->session->set( 'shipping_for_package_' . $package_key, false );
        }
    }
}

Add optionally this code functions to your main plugin file.

Tested and works, Updating the shipping cost.

Important note: Don’t forget to empty your cart to refresh shipping method caches on first start.

Related:

• Refresh cached shipping methods on checkout update ajax event in Woocommerce
• Disable shipping method based on chosen payment method in Woocommerce