Now, if your hypothetical situation is a couple saying “We were married on such and such a date, at this parish” but the pastor responds with “There is no record of it. I consider you to be in a state of fornication.” Then we are dealing with a matter of declaring a juridic fact–the married state, or not, of the parties. If the people involved could not come to an amicable solution on their own, then the couple could approach a tribunal. The Judge should try to have the people settle the matter outside of court (cf. c. 1446) but if that fails and the couple presents a petition, I think the judge would have to accept it and start the case.
The actual process used would not have to be as lengthy as a marriage nullity case but it would still involve a petition, citation of respondent (the priest), formulation of the doubt, gathering of evidence, and decision. This is the “oral contentious process” (cc. 1656-1670). If it should happen that the judge determines that the couple did, in fact, marry in accord with canonical form at the parish, I contend that he could not order the priest to do anything. The judge would only declare that the couple is presumptively married. They could then go to the bishop and the bishop would have the authority to command that the marriage be properly recorded by the priest.
If, on the other hand, it is a matter of a couple saying “We were married on this date, at this location but there is no record of it” but the priest says “Fine, you’re married. But, I don’t care about the records” then this is (in my opinion) a matter that the judge has no competence over. The couple could not go through the judicial branch. The executive branch of government is competent (the chancery, up to the bishop) to get the records corrected.