Are Catholic priests allowed by church law to provide professional counseling and referrals to penitents who come to them to confess? Let us say we have a priest, who is a licensed psychiatrist as well. A devout Catholic confesses their sins to said priest. The Catholic person chats with the priest, and priest asks follow-up questions like a doctor would. Priest then determines that the person is suffering from a psychological disorder, in this example, depression.

From a professional/moral standpoint the penitent/"patient" should not be obligated to pay the priest for his profession, since they came to confess, not to seek medical help. But as the priest orders penance, is the priest allowed by Church law to disclose to the penitent his diagnosis, offer counseling, and/or suggest to the latter to at least seek another psychiatrist for a second opinion?
In essence, care should be taken lest the Sacrament of Confession be confused with psychological counseling. A priest with a degree in psychology should be sure not to confuse the spiritual purpose of the Sacrament with the therapeutic purpose of psychology.

Therefore, the proper procedure would be for the priest to instruct the penitent to bring up his psychological issues outside of Confession. In Confession, what is dealt with is sin and grace (see Catechism of the Catholic Church [CCC] 1422); psychological counseling regards the cognitive and emotive faculties.

Naturally, if the priest is acting in the capacity of a licensed professional psychologist (outside of Confession, obviously), then he may seek remuneration like any other. He may not, of course, charge for Confession. (See CCC 2121.)

(I do not believe there is any official legislation regarding this issue, but I suggest reading the overview on the Sacrament of Penance in CCC 1422-1470; you will see that psychological therapy does not really belong to it.)

Aside from the theological issues, confusing Confession and counseling poses practical issues: for example, a priest can never divulge anything he has learned in Confession for any reason. (See Code of Canon Law, 983.) Hence, for example, it would be complicated for him (if not impossible) to consult other psychologists, or refer the “patient” to other psychologists.

The proper procedure, then, would be to counsel the penitent to seek psychological counseling (should that be advisable). If the priest is himself a licensed psychologist, he could even suggest that the penitent seek his help outside of confession.

(I think that most priests in this situation, however, would advise the penitent to go to a different therapist or counselor, since it can be complicated to keep track of what has been said under the seal of Confession and what has not.)