If you just want to update your site extension module to the latest version, you actually do not need to manually update it through the backend, because when you update the extension to the next new release version using the latest launcher or full installer, the installation page there have an options to select whether you need to update all the modules.

So you do not need to switch your license domain from the live site to the test site.

By the way, your Easysocial modules are running the latest version already.