Editing Core files in the right way?

So, I need to edit the wc_cart_totals_shipping_html function in the wc-cart-functions.php file located in the includes folder.

I can’t, for the life of me, figure out how to properly override this function. It doesn’t seem to have filters or actions attached to it in anyway… so what do I do? Anyone have any suggestions at all? Basically, I need to send custom values through the ajax when someone clicks a shipping option and sending the data through the system.

I essentially need to add values to the “wc_get_template” and pass through the variables

function wc_cart_totals_shipping_html() {
	$packages           = WC()->shipping->get_packages();
	$first              = true;

	foreach ( $packages as $i => $package ) {
		$chosen_method = isset( WC()->session->chosen_shipping_methods[ $i ] ) ? WC()->session->chosen_shipping_methods[ $i ] : '';
		$product_names = array();

		if ( count( $packages ) > 1 ) {
			foreach ( $package['contents'] as $item_id => $values ) {
				$product_names[ $item_id ] = $values['data']->get_name() . ' ×' . $values['quantity'];
			}
			$product_names = apply_filters( 'woocommerce_shipping_package_details_array', $product_names, $package );
		}

		wc_get_template(
			'cart/cart-shipping.php', array(
				'package'                  => $package,
				'available_methods'        => $package['rates'],
				'show_package_details'     => count( $packages ) > 1,
				'show_shipping_calculator' => is_cart() && $first,
				'package_details'          => implode( ', ', $product_names ),
				/* translators: %d: shipping package number */
				'package_name'             => apply_filters( 'woocommerce_shipping_package_name', ( ( $i + 1 ) > 1 ) ? sprintf( _x( 'Shipping %d', 'shipping packages', 'woocommerce' ), ( $i + 1 ) ) : _x( 'Shipping', 'shipping packages', 'woocommerce' ), $i, $package ),
				'index'                    => $i,
				'chosen_method'            => $chosen_method,
				'formatted_destination'    => WC()->countries->get_formatted_address( $package['destination'], ', ' ),
				'has_calculated_shipping'  => WC()->customer->has_calculated_shipping(),
			)
		);

		$first = false;
	}
}

Many thanks for all who can maybe help me.