I am very sorry to tell you that her condition is very common in hens over 2 1/2 years old, if they are production hens. I have lost five of my hens with this. They lay so many eggs in such a short time that their bodies wear out. Their abdomen gets distended because of fluid build up, they get an infection and start to deteriorate (loosing a lot of weight). If the fluid is not taken out (by a vet) it just continues to build up and even if it gets taken out, it will build up again so it is a lost battle.
There will be a point when you will have to decide to let her go because her quality of life is not very good, neither is her prognosis.
I spent tons of time and TLC with my first hen that had this condition until the battle was lost and I had to have her euthanized by the vet. Now I have come to recognize the symptoms and make decisions as to when it is time to let them go. Also I do not get hatchery chicks anymore, I get heritage breeds from local breeders, I still have 3 of my original hatchery hens, they look and act healthy so I hope they don't get the same illness as their sisters. This is a very heart breaking situation because there isn't anything I hate the most than seeing an animal suffer.
I had a necropsy done in 2 of these hens and they had ovarian cancer (tumors all over the reproductive tract).
I am not saying that this is definitely what your BR hen has but according to your description it sounds very much like it.
I wish you the best luck