If she were egg bound then she would likely already be gone by now (and she would not be pooping if egg bound). It's far more likely that she has a reproductive problem, it is not uncommon in hens starting around the age of two, especially in high egg producing breeds, which an ex-bat would be. Common issues can be internal laying, salpingitis, and reproductive cancers. Her body condition suggests that it has been going on for a while, it just may not have been obvious, they are very, very good at hiding illness. They can have good days, and not so good days, but generally it will not get better. Once the abdomen is bloated like that there is usually not much that can be done for them really, it indicates that whatever the issue is, it's advanced. The long term prognosis is poor. I'm sorry, I know that is not what you want to hear. A clear and definite answer is often not known until a necropsy is done, symptoms can be very similar for the various reproductive problems. When I have a hen like this I leave her with the flock as long as she is acting like a normal chicken. Once she does not, acts obviously unwell, goes off food and water, isolates herself, or is drawing attacks from the flock, then I don't let her suffer.
