Of course, you could. You should go in the Tools tab in ribbon, click on Backup and follow the instructions. You can also restore backups you've previously made anytime later by clicking on Restore button in Tools tab in ribbon.
Embedding the key directly in GameXT would lead to that each user uses the same key and thus maximizes the chances to reach the API limits. With your own GiantBomb API key, you are sure you are the only one unsing it (it is tied to your GiantBomb account) and it will be harder for you alone to reach the limits.