I want to know if it is possible to asynchronously invoke a Yii controller method from one of its actions while the action renders a view, leaving the method
Typically, the solution for these kind of problems would be to integrate a message-bus in your system. You could consider a product like Beanstalkd. This requires installing software on your server. I suppose this suggestion would be called "using an external library".
If you can access the deployment server and you can add cronjob (or maybe a sysadmin can) you could consider a cronjob that does a php-cli call to a script that reads jobs from a job queue in your database which is filled by the controller method.
If you cannot install software on the server you're running, you could consider using a SAAS solution like Iron.io to host the bus functionality for you. Iron.io is using what is called a push queue. With a push queue the message bus actively performs a request (push) to the registered listeners with the message content. This might work since it doesn't require you to do a curl request.
If none of the above is possible, your hands are tied. Another post which is quite relevant on the subject: Scalable, Delayed PHP Processing