Detraction is the unjust damaging of another's good name by the revelation of some fault or crime of which that other is really guilty or at any rate is seriously believed to be guilty by the defamer.

An important difference between detraction and calumny is at once apparent. The calumniator says what he knows to be false, whilst the detractor narrates what he at least honestly thinks is true. Detraction in a general sense is a mortal sin, as being a violation of the virtue not only of charity but also of justice. It is obvious, however, that the subject-matter of the accusation may be so inconspicuous or, everything considered, so little capable of doing serious hurt that the guilt is not assumed to be more than venial. The same judgment is to be given when, as not unfrequently happens, there has been little or no advertence to the harm that is being done.

The determination of the degree of sinfulness of detraction is in general to be gathered from the consideration of the amount of harm the defamatory utterance is calculated to work. In order to adequately measure the seriousness of the damage wrought, due regard must be had not only to the imputation itself but also to the character of the person by whom and against whom the charge is made. That is, we must take into account not only the greater or lesser criminality of the thing alleged but also the more or less distinguished reputation of the detractor for trustworthiness, as well as the more or less notable dignity or estimation of the person whose good name has been assailed. Thus it is conceivable that a relatively small defect alleged against a person of eminent station, such as a bishop, might seriously tarnish his good name and be a mortal sin, whilst an offence of considerable magnitude attributed to an individual of a class in which such things frequently happen might constitute only a venial sin, such as, for instance, to say that a common sailor had been drunk. It is worthy of note that the manifestation of even inculpable defects may be a real defamation, such as to charge a person with gross ignorance, etc. When this is done in such circuмstances as to bring upon the person so disparaged a more than ordinary measure of disgrace, or perhaps seriously prejudice him, the sin may even be a grievous one.

There are times, nevertheless, when one may lawfully make known the offense of another even though as a consequence the trust hitherto reposed in him be rudely shaken or shattered. If a person's misdoing is public in the sense that sentence has been passed by the competent legal tribunal or that it is already notorious, for instance, in a city, then in the first case it may licitly be referred to in any place; in the second, within the limits of the town, or even elsewhere, unless in either instance the offender in the lapse of time should have entirely reformed or his delinquency been quite forgotten. When, however, knowledge of the happening is possessed only by the members of a particular community or society, such as a college or monastery and the like, it would not be lawful to publish the fact to others than those belonging to such a body. Finally, even when the sin is in no sense public, it may still be divulged without contravening the virtues of justice or charity whenever such a course is for the common weal or is esteemed to make for the good of the narrator, of his listeners, or even of the culprit. The right which the latter has to an assumed good name is extinguished in the presence of the benefit which may be conferred in this way.
 

It depends, but no, you cannot refuse. You can say that you prefer not to name him, but if the priest insists (which he shouldn't), you have to tell him, otherwise you are not obligated to name him, more like your first obligation is to not name him.

Me too.  It seems to me that mentioning the name (whether on one's own or because the priest asks you to mention it) would only add to the sin.  Now this particular priest is not only being bad-mouthed outside of confession, his good name is being dragged into confession. Something seems off to me about it, and that is why I doubt a priest would ask for the name.

I agree, but it's up to the priest to decide what is essential and what is not. If the priest needs to know the name in order to be clear on the sin he is forgiving, then he might ask. Which is a good reason to be blunt when confessing one's sins, to try to leave no need for the priest to ask such a question. Either way, no need for the penitent to offer the name at all unless the priest insists on it.

I’m not so sure of this. I believe you can refuse since knowing the name of the priest is not absolutely essential to the type of sin committed.

Speaking bad about a priest is one thing to confess, but I think the priest may ask in what way have you spoken badly about that priest? Via word of mouth to others? In an public forum like on the Internet for the entire world to see? I'm thinking more about the restitution that needs to be done in reparation.

I also do not think the priest will ask for the name as it is not essential.