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?
Gryffon Vom Wildhaus
BH, OB1, TR1, AD
Rottweiler/Hairy Dog mix?? 2004-2015