I don’t understand what you mean by “it is treated like a mixed marriage”? There is no permission from the bishop required, right? Or is there?
Yes, there is.
Can. 1071 §1.4 Except in a case of necessity, a person is not to assist without the permission of the local ordinary at a marriage of a person who has notoriously rejected the Catholic faith;
§2. The local ordinary is not to grant permission to assist at the marriage of a person who has notoriously rejected the Catholic faith unless the norms mentioned in can. 1125 have been observed with necessary adaptation.
Can. 1125 The local ordinary can grant a permission of this kind if there is a just and reasonable cause. He is not to grant it unless the following conditions have been fulfilled:
1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;
2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;
3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.
Besides, neither one is practicing. Is that a different case?
When I responded to his question, I did not know neither one was practicing. It sounded like he was non-practicing and she wanted to practice. He makes that clear in a late post. In that case, they need to speak to their pastor. The pastor will need to counsel them. If neither intends to practice, it is unlikely the pastor will proceed with the sacrament, it might be postponed.
Marriage is a natural right, so I could see why a priest wouldn’t do anything but heavily encourage that they start practicing, but yet agree to marry them.
It’s a natural right, but not an unlimited one. If they can’t validly enter a marriage, he cannot proceed.
As for knowledge and will, it seems clear that the OP and his fiance were well aware that sexual relations outside of marriage are morally forbidden, which I think is the sin the OP was refering to.
But they may not have been aware that they were not considered married.