If you activate the licence on the testing domain, you will need to deactivate it before going live, then activate it on the live domain. You can also manage this in Your Account.
The plugin will work without the licence being activated, as the purpose of the activation is to allow automatic updates in the WordPress plugins page. No other functionality is impacted by not activating the licence. In that case, you can download the latest (newer) version as a zip file for Your Account, and, if you have WordPress5.5+ you can add the new version using the Add New Plugin -> Upload option in WordPress, even though a previous version is still installed.