A judge may take into consideration the letter from a physchiatrist but the mother would need to file for modification of custody to take away visitations in a court order. She would then have to convince the judge/court that it’s in the child’s best interest NOT to visit with the other parent and from what you posted, there’s not enough for a judge to do that. Since she has been held in contempt before it may be that the judge would not even hear the case. Make sure to keep documentation of everything. Record the phone calls between her and your fiancee. Have him let the ex know that he will start monitoring this situation with the calls and if the child begins to get upset he will disconnect and she will not be allowed to speak to the child again until the next scheduled phone call. Put it in writing. He doesn’t have to listen in on the actual call to know if the child is getting upset.
The doctor should meet with both parents individually or together to discuss these issues. I suggest that you wait to see if there is an actual letter sent or if she’s just trying to cause you to worry needlessly. If you get a letter, contact the doctor to set up an appointment to discuss these issues. Document everything.
I do not know how old this child is but it could help to sit the child down when he/she gets upset about this and explain that you miss and love them also. The mother obviously knows that the child is going to be upset and is doing her best to make these visits worse on the child. It’s not uncommon for this to happen.