It is easy to be anonymous but that is not what you are asking for: You want users to be firstly identifiable as belonging to a set of non-anonymous users. That is what makes it hard to guarantee anonymity when an audit trail already exists.
What you want is something like an arms length relationship between the two states of identification and anonymity. You were trying to provide both from the same system when the confirmation that someone belongs to the organisation must happen totally independently from Discourse.