Hey Svan, damn it sucks you are getting stuck by this! Sorry!
1. Did you try posting on the NFC Ring forum?
RE "3-party app" our app never stores your details, we use the Google provided Google Auth libraries to handle the authentication process. I can personally guarantee and promise at no point do we store or use your username/pass, we just get a success/failure callback from them and act accordingly.
2. Is it possible you can use your Google Auth or is your work-around preventing you from doing this?
We're actually just beta testing an updated version of the app that forces users to enter a pin code to handle circumstances such as this. Flight mode and two-form factor auth have been an issue for us so we're taking pro-active steps to address it ASAP
This doesn't help you though ;(
3. As BabelHuber asked, do you have debug mode available? If so a simple adb uninstall of the app will get you moving again.
Sorry about the delay through support, I just took another team member on today to help with the backlog, he starts tomorrow so hopefully things will move faster!
Source: I'm the inventor of the NFC Ring