I have just finished a series of cognitive behavioural therapy sessions. Before they started, I had an appointment with a psychiatrist whose report went to the therapy manager - and it was their responsibility to pair me up with someone they felt I would work well with.
Therapy is an intense and personal relationship - if there is stress and tension in the relationship between counsellor and client, it will not work, or it won't work well.
IMO, the counsellor did the right thing when he said that he wouldn't feel comfortable with her feeding during the sessions - please note, I am NOT saying he is right to feel uncomfortable around breastfeeding mothers, but as he does, it would negatively affect the counselling relationship.
However, the service should have a method for dealing with this issue - it would seem reasonable for her to go back onto the waiting list, to be placed with the next counsellor who was comfortable with counselling a breastfeeding mother. It cannot be the first time the service has dealt with a similar issue - both counsellor and client are human beings, and sometimes the relationship will just not click - so if the counselling is to be effective, there must be some mechanism for allowing the client to move to a different counsellor, or the counsellor to say that they feel the relationship is not working and that the client would be better off with a different counsellor.