I am sorry to hear you are worried about your Bulldog. My name is Rebecca. I am a veterinarian and will do my best to help. She sounds like she was a very sick dog and I am glad to hear she pulled through her illness.
So she has been incontinent for 3 months? That must be difficult for her as well as you and your family.
Does she have diarrhea or is her stool normal now? How is her appetite?
Were any tests (Xrays, MRI, consult with a neurologist) done to determine exactly why she is incontinent? That is not a normal consequence of a bad bout of pancreatitis and hemorrhagic gastroenteritis. Did she suffer nerve damage? Is there no hope of her recovering function? Were any treatments tried for the incontinence such as proin (urinary incontinence), prednisone, hormones?
I agree it must be hard for her to adjust to her new life. Most dogs eventually learn to accept wearing diapers; how long have you been using the Depends?
All I can tell you is what I would do if this were my own dear dog in this situation. I personally would not be contemplating euthanasia at this point if she is eating and getting around well. It would be sad for me to see her being affected by being confined, and wearing diapers, but to me personally, that is not reason enough to part with her forever. I would give her extra attention and treats, try to train her to let me know when the diaper needs to be changed, and spend as much "quality time" with her as my busy schedule permits. But what is right for me is not necessarily right for every dog owner. You have to discuss matters with your family, and make a decision that is best for all concerned. The worst thing is if the dog starts to feel resented or unloved.
This is a hard one. These are just my first thoughts. Let me know if this helps at all, and what other questions you have.