In theory, you could go to the diocese in which you live. One's standing with the Church, so far as I am aware, has no bearing on whether they will make judgment on an annulment or not. In theory, a non-Catholic, or even a non-Christian, could go to the Church and ask for a judgment on their marriage or marriages. All an annulment does is to say, in the eyes of the Catholic Church (and, as I always tell my son, thus in the Eyes of God), that a valid marriage does not exist.
For lack of canonical form annulments, so far as I am aware, pre-Vatican II and post-Vatican II norms are identical. (I will stand corrected.) I can't see that being a problem.
As far as often subtle and nebulous subjective psychological factors, obviously the traditional approach and the post-Vatican II approach are far different. If one received an annulment on such basis, it would be up to their own conscience, whether to accept this judgment, or to say "these wouldn't have been traditional grounds, I cannot accept this annulment".
As far as the possibility of doing so outside of the diocesan structures, so far as I am aware, there is no way to have a marriage declared null. I do know that, supposedly, Hutton Gibson declared Mel's marriage to Robyn invalid. He had no authority to do this.