To marry a non-baptized person is called "lack of canonical form."
Disparity of cult is when there is different faiths but assumes that both are practiced. One of who can still be unbaptized.
Either can be dispensed by the bishop but neither can have a nuptial Mass as far as I know. Why would it matter after that. A justice of the piece can do the same thing.
In effect, the Priest is witness the same as a justice of the piece in that case.
In either circumstance, if the non-Catholic party is not baptized, would hold any sacramental form and be non binding by the Church unless proper dispensation were granted but under what circumstances that would even be requested I am unaware.
The marriage is illicit though made valid by the civil contract, not by the sacrament because canonically the sacrament cannot be conferred by the unbeliever.
Basically, what you have is a marriage outside of the Church with a priest as a witness instead of some other non-catholic official. A Catholic priest officiating does not a Catholic marriage make.
Thus, if there was ever a civil divorce, no annulment would even be necessary because impediment of form was absent at the ceremony. Only a declaration of lack of canonical form would be needed to free the Catholic to attempt a marriage.
Though if it lasts, then its possible for the believer to be the salvation of the non believer. Yet if the non-believer should choose to leave, then the believer is free to attempt a true marriage with nothing more than a declaration of lack of canonical form.
The questions to ask would be, why bother with a priest or the Church at all or why would the priest bother officiating other than to catechise the non believer who would be reguired to agree to raise any offspring Catholic and or protect the interest of the Catholic.
Is that clear as mud?