Isn't the old adage that possession is 9/10 of the law? I would send her a cease and desist letter, informing her that you do not want contact with her in any way, shape or form from this date forward. Inform her that any further contact will be through your attorney. When she realizes that contact will cost her money, she will change her mind., she doesn't want the dog, she just doesn't want you having it. It is not fair to the dog to be uprooted and she has no legal standing, in my opinion, because she a) abandoned the dog, b) has had no contact with the dog, c) has not participated in the expenses incurred from having the dog (to include feed, vets, licensing etc) d) this dog has been assimilated into your family, to include your children and it would be cruel and unusual punishment to have the dog taken from them. Then serve her with a no contact order for you, your wife and property. This is not to be construed as legal advice, consult an attorney if it means that much to her. I would not lie to her, just tell her to get bent, sue me for custody and I will see you in court. Being deployed, she could not have taken care of the dog anyway.
good luck,
Scott