Congratulations on your wonderful desire to donate.
Each transplant hospital has its own evaluation process. The mental health of each potential donor is taken into consideration along with many other things. I think that they try to evaluate each person, and don't make a blanket refusal of all persons who are suffering or have suffered from a mental problem, and they do take into account whether the problem is now under good control. But they do have to seriously consider:
--- Does the person's mental state enable him/her to make a free and fully informed decision to donate? For example, if the person is currently experiencing wild mood swings he/she might offer to donate while in a very optimistic and excited mood, but then regret the offer when a big drop in mood occurs. So they have to be sure that the person's condition is now stable enough to make a free decision.
--- A notable percentage of living organ donors experience depression after donation, even if the donation is successful. So if the potential donor has had a history of depression the danger of post-op depression has to be taken into consideration.
--- If the potential donor is taking medication for his/her problem, the hospital has to evaluate possible negative reactions between those medications and the medications (anesthesia, pain-killers, anti-inflamatories) that the donor will have to receive.
So it is best to be totally honest with the hospital about one's physical and mental health history, and be prepared to argue your case reasonably. They will probably want to have a written evaluation and recommendation from your current health care provider.
best wishes,
Fr. Pat