Solution would be to add an action log entry with the ID and title of the topic, when an anonymous post by a logged in member is made.

Hardly anonymous if you do this. Sounds like you don't want to offer anonymous posts to me, or you are going to do so misleadingly. Surely you can moderate the posts themselves if not the member without any additional "tweaks". You should at least call them something like confidential posts if you make this change as they will no longer be anonymous.

Nothing is anonymous technically. But the reason I feel it is important staff know who posted anonymous posts is for moderation, security, and ultimately liability / legal. If someone posted an anonymous post, say, exposing the identity of one of the other member, it is vital to be able to easily identify the post to its member by staff to enforce the law and rules.

The purpose of anonymous posting is, or should be, to be anonymous from other regular users... Not staff.