As I told here, we proceeded with our task without this member in question and informed them about it as well. One major factor to the decision was that they clearly informed us they won't be reading their email for a while, which pretty much sealed that we consider them impossible to work with as the deadline is next week and they don't respond to questions. Which we told to the professor in charge when we informed them about a change in our group size.
But guess what?
That member actually wrote a casual email to our professor today with general stuff about our essay, signing it with "on behalf of all", like nothing has changed. (So apparently they ARE reading their email.)
I guess this just emphasizes the fact our communication does not work, but at the same time I think I have all the reason to be thankful for people in my past, because they trained me pretty well to deal with e.g. talking with walls and pathological liars... just that I never thought that kind of people would actually get this far in their studies.