Dev Note: Needs improving. Possibly a breakdown.

​
Creating a new callback

Copy and paste this example class to get you started. Remember to label the callback class, name and set the parameters you wish to pass in the request.
Callback Class Example
<?php
/**
 * A custom callback for CoCart triggered when updating the cart.
 */
if ( ! defined( 'ABSPATH' ) ) {
	exit;
}

/**
 * Callback example class.
 */
class My_Custom_CoCart_Callback extends CoCart_REST_Callback  {

	/**
	 * Callback name.
	 *
	 * @access protected
	 *
	 * @var string
	 */
	protected $name = 'my-custom-callback';

	/**
	 * Callback to update the cart.
	 *
	 * @throws CoCart_Data_Exception Exception if invalid data is detected.
	 *
	 * @access public
	 *
	 * @param WP_REST_Request $request    The request object.
	 * @param object          $controller The cart controller.
	 *
	 * @return bool Returns true.
	 */
	public function callback( $request, $controller ) {
		try {
			if ( $controller->is_completely_empty() ) {
				throw new CoCart_Data_Exception( 'cocart_cart_empty', __( 'Cart is empty. Please add items to cart first.', 'cart-rest-api-for-woocommerce' ), 404 );
			}

			$data = isset( $request['data'] ) ? $request['data'] : array(); // Custom parameter.

			$cart_updated = false;

			if ( ! empty( $data ) ) {
				// Place your callback here.

				$cart_updated = true;
			}

			if ( $cart_updated ) {
				// $controller->calculate_totals( $request, true ); // Uncomment if you need to recalculate the cart totals.

				// Only returns success notice if there are no error notices. - Feel free to change the message response.
				if ( 0 === wc_notice_count( 'error' ) ) {
					wc_add_notice( __( 'Cart updated.', 'cart-rest-api-for-woocommerce' ), 'success' );
				}
			}

			return true;
		} catch ( CoCart_Data_Exception $e ) {
			return CoCart_Response::get_error_response( $e->getErrorCode(), $e->getMessage(), $e->getCode(), $e->getAdditionalData() );
		}
	}
} // END class

return new My_Custom_CoCart_Callback();

​
Registering the callback

Simply load the class file created and hook the callback like so.
If you registered a callback before with version 4 or lower of CoCart then you will need to register them again. We simplified the callback system to perform better in version 5.
add_filter( 'cocart_rest_callbacks', 'register_my_custom_callback' );

function register_my_custom_callback( $callbacks ) {
	include_once( dirname( __FILE__) . '/callback/my-custom-callback.php' );

	$callbacks[] = new My_Custom_CoCart_Callback();

	return $callbacks;
}

​
Calling the callback.

E.g. Triggering the callback to use 10 points to save on the cart total. 
curl --location --request POST 'https://example-store.com/wp-json/cocart/v2/cart/update' \
--header 'Content-Type: application/json' \
--data '{
    "namespace": "my-custom-callback",
    "data": {
      "points": "10",
    }
}'