My web application is deployed to a production site, but I also need to perform local development. Besides, I have a couple of test servers where I perform different testing
You will have to instead create different application for different Use Cases as per your requirement. This is one strange limitation that you will have to deal with.
Although unrelated but Heroku's (which is partner Cloud service provider) Getting Started Guide for Facebook suggests the same work through for working locally with the application along with development.