I ran into simmilar issues with aurora a while ago. From my understanding this is not related to LOS or MicroG. It's just Google blocking the anonymous accounts used by Aurora. What others suggested and I successfully implemented is creating a dummy google account with no personal info, and use it to with Aurora.
A few notes on this. If they ask for a phone number for verification use some online service and immediately add fallback options for login (like an authenticator app - i recommend Aegis). My account got blocked after a few day, but i successfully recovered it aferwardas. Can be a bit of a hassle to setup, but it's been working without issues since then.