Render the Prestashop's blockcart module in Wordpress (or any external script)

prestashop-cart.php

<?php

// Prestashop's root folder, in my example Prestashop is installed in the root folder, and Wordpress is installed in /blog
define('PRESTASHOP_ROOT', ABSPATH.'..');


// Still don't know if necessary
// require PRESTASHOP_ROOT.'/config/settings.inc.php';
// require PRESTASHOP_ROOT.'/config/defines.inc.php';

// Essential Prestashop require to bootstrap and load dependencies
require PRESTASHOP_ROOT.'/config/config.inc.php';
// Load the desired module to render
require PRESTASHOP_ROOT.'/modules/blockcart/blockcart.php';

// Nope. This seems to load the entire Prestashop and throw a 404
// Dispatcher::getInstance()->dispatch();

// Seems like it is required to get the context filled
if (isset(Context::getContext()->controller)) {
  $controller = Context::getContext()->controller;
} else {
  $controller = new FrontController();
  $controller->init();
}

// Get the cookie and the cart from context (we need params to call the hook)
$context = Context::getContext();

$hook_args = [
  'cookie' => $context->cookie,
  'cart' 	 => $context->cart
];

// Initialize the module
$blockcart = new BlockCart();
// Display the top hook (rendering Smarty, etc)
echo $blockcart->hookTop($hook_args);