Peter,
There are two options:
1. If you use the existing app for your new domain, you can use Autotune to set a number of settings (which may break your existing site). However, there are some fields such as OAuth Redirect URIs that are required for Facebook Login, but not accessible through FB's API to us. You'd have to set these manually. If you went this route, the existing site's would not be configured properly.
2. The safest way so that you don't break existing site functionality is to create a new FB application. You can use our
setup guide
to get started for a new app. But the easiest may be just to go through tab by tab of the FB app settings. It really shouldn't take more than 10 minutes to get a new app set up if you're just copying from an existing one.
It's really a matter of if you need the existing site to continue working while you're testing the staging server.
-Melissa