I just solved the same problem. After a long time when I was not using BC I restarted it and apparently an updated version of the application displayed a strange looking window where I was asked to log in to the service. Subsequently, the MFA code from the authenticator app was rejected about 5 times, fortunately the last one was accepted. All I was shown was exactly what you were describing, basically management of BC account. I checked out, quit the app and tried again, then unfortunately I didn't go through MFA.
So I went to the web and logged in (here MFA worked for the first time), removed the MFA login preference (impossible to use it if such unreliable), restarted the BC application and again without success.
I downloaded the installer for Windows from the web, ran it (he didn't want to fix anything, just a new installation took place) and then I logged in again. I was told that I exceeded the number of licenses (even though it was the same computer), so again on the web, delete the original record of the device and then everything worked, it seems that it even took over the original preferences.
So my story had a good ending, but it cost me 30 minutes of time to demonstrate again how naive it is to rely on cloud services where we have nothing under control.
I wish you good luck - in your place I would try to uninstall everything, check that it is really gone, restart and then try to install again.
EDIT: Maybe just try to go to web portal, delete this device record and try to restart application.
