Does it work for other sites that use 2FA?
I've had a mooch round the Mozilla support sites, and it may be that you need to disable advanced protection for BC, as its possible that the cookie confirming the authentication is not permitted.
https://support.mozilla.org/en-US/kb/total-cookie-protection-and-website-breakage-faq#w_what-kind-of-breakages-does-total-cookie-protection-cause
I can install FF and test, but probably not this weekend.