Yup. No good answer. If the servers are down you will be unable to log back in and Bluetooth key will be disabled.
I would only suggest logout/login if you have the key card in hand.
mine was also spinning this morning. It was set to my S. It also wouldn’t let me swipe to my 3. I clicked the gear, selected my 3 (stumbled on it, didn’t know that option was there in the gear menu). The 3 loaded and then I was able to select back to the S and it worked too.
I guess the solution is to buy a second Tesla.