Hi Welcome to the forum, and I'm sorry that you having to deal with serious health issues. Hope you are getting well.
Now for Brody, I'm confused what the deep-seated behavioral issues are? So far, unless I'm missing something, we have:
a) diarrhea, and
b) dog destroyed dog bed.
I think I'm missing something, because it seems that you have found a good solution (keep her separated from the other dogs, with a kong to keep her busy), and you have increased her daily exercise. - exactly what I would have suggested for a dog that suddenly started to be destructive.
I don't see how any of this is grounds for euthanasia? If you are worried that she may harm the other dogs, then doing what you are doing now, or finding her a new home would be workable solutions, and should not be a death sentence.
Another question is what are your goals for the next three months? What behaviours are you wanting to change and how do you plan to go about that?