You mention that you need to keep your Fauna key safe, which I agree with. However, you then embed those keys into your app. Those keys will ship as part of your JS files.
Since the keys you created in this example are admin keys I'm guessing that would give someone full access to your data. You might be ok with some restricted read-only keys. Otherwise, you may want to move your data access to something like a serverless function.
The Vue CLI documentation specifically says not to store private or secret information in the .env files.