If she cares as much about Izzy as she says she does, I think she will find a better way to deal with this. I really believe RR is a true lover of animals. Her schedule may not permit her to spend as much time with her as she needs to but I hope this brings out a need to make some time in one or both of their schedules. To put the dog down would be ridiculous without trying to train her or muzzle her or whatever else they can do.
Best wishes to all of them. I would not like to hear about euthanizing her for "convenience". She seems like a good girl and deserves better.