I really don't like option 3, I'm okay with option 4, but I prefer option 1. It just makes sense to me, considering how I believe you want this forum to be, that invite codes are given away in a case-by-case basis, since, unlike 2, you'll at least be able to know that the new users have been recommended by an existing one.
2 would basically mean opening it to the public literally, like Fristi said