Google are terrible at developer support, but I think it is fair to say that they didn't foresee anybody using the Tasks API in this way, where you are proxying for all your users.
Keep persisting, but as a temporary measure I would prompt your users to provide you with their own API key. Lots of apps do this (such as CMS plugins). You can give them a link to the API portal and explain in 2-3 points how to generate a key and paste it back in your app.
Also, your app is in debug mode, so errors are showing a complete stack trace. Your database is also down right now.