As far as I know, priests aren’t allowed to give “dispensations” on anything. That power would belong to the Holy See. It sounds as if this priest acted on his own authority.
The Church does NOT permit the use of artificial birth control for any reason.
The Church does not permit sterilization procedures, i.e. medical procedures that are performed for the purpose of preventing conception. It does permit medical or surgical procedures that are performed for other reasons of health that happen to cause infertility but it does NOT permit procedures that performed for the specific reason of preventing conception regardless of how “noble” the reasons for these might be. This couple could have used NFP, which is quite effective.
The problem with allowing married couples to use artificial birth control is that it opens the door to the widespread use of birth control. Indeed, Catholics who wanted the Church to approve the use of the Pill in 1968, wanted married, faithful Catholics to have the option of using the Pill for supposedly good reasons, they didn’t intend for everyone to be able to use it. Of course, Pope Paul VI quashed their hopes in Humanae Vitae and unequivocally condemned the use of the Pill or any other form of contraception by Catholics for any reason. Many Catholics simply refused to accept this teaching and, not surprisingly, many Catholics, married and single, started using artificial birth control, some for “good” reasons and some for bad. At least those who use birth control, know that they are disobeying the Church and doing something wrong.