Birth control as in 'no baby chickens/fertile eggs' is easy - just keep the roo away.
But in your case what I think you are really talking about is keeping the hen from laying/producing eggs at all, so as not to aggrevate or cause further prolapsing.
There are only two ways I know to stop a hen from laying completely...
One is the chicken version of a hysterectomy - removal of the egg producing organs through surgery. Not a cheap option, but will work, and the ONLY cure for chronic prolapsing with egg laying OR internal laying. You'd have to find an avian vet and/or vet school, etc that would be willing to do the surgery - a 'regular' vet, and even many farm/livestock vets would probably not do the surgery due to the complexity, their lack of familiarity/skill in the procedure, and the dangers of anesthesia in birds.
The other is cheaper but maybe not what you were looking for - it's to euthanize the hen.
You might also want to consider contacting any avian vets in your area - other options they they know of for other bird species might work for chickens....
But in your case what I think you are really talking about is keeping the hen from laying/producing eggs at all, so as not to aggrevate or cause further prolapsing.
There are only two ways I know to stop a hen from laying completely...
One is the chicken version of a hysterectomy - removal of the egg producing organs through surgery. Not a cheap option, but will work, and the ONLY cure for chronic prolapsing with egg laying OR internal laying. You'd have to find an avian vet and/or vet school, etc that would be willing to do the surgery - a 'regular' vet, and even many farm/livestock vets would probably not do the surgery due to the complexity, their lack of familiarity/skill in the procedure, and the dangers of anesthesia in birds.
The other is cheaper but maybe not what you were looking for - it's to euthanize the hen.
You might also want to consider contacting any avian vets in your area - other options they they know of for other bird species might work for chickens....