How can I get a developer license for my staging system?

First of all: You won't need to purchase an extra license!

A developer license can be generated via My Atlassian. After logging in, you will be directed to the license area. There you open the product you need the developer license for. You will gain some more information on the product including the area with your current license key. Underneath this area you find the actions menu:

Choose View Developer License and the license for your staging system will be displayed in a pop-up window.

 

You must be one of the technical contacts of the license. Otherwise you won't see all options. If you are not a technical contact, ask one of the displayed technical contacts, if they could add you or generate the license.