People often are too soft on adopted dogs. Or are afraid of them once they get them home and there is no one from the shelter handling the dog for you.
I know a lot of people are also into rewarding good behaviors only (positive training). I believe a dog needs both the positive when displaying good behavior, and corrections when they are being butt heads. Whether that be a firm vocal no, remove the dog from the situation in a “time out,” or a physical correction if the behavior warrants it. I train with a balance of positive training, and corrective training.
How long have you had the dog?
I’d start out with ignoring the dog altogether when he displays this behavior. Not just your wife, but also you. No attention besides feeding and bathroom breaks, and keep those all business. When he/she seeks attention, tell him to go lay down. He hasn’t earned your attention yet. After a day or two, if he is still displaying this behavior, get into his space physically while giving him the back command. Keeping walking him back until he looks away or moves away on his own. This one will likely cause some disagreement on the forum. I’m not a professional trainer, just stating what has worked with my fosters and rescues.
Don’t amp the dog up with treats and training before your wife gets home. That’s counterproductive. He’s in a heightened state when you do that, which makes him more aware of your wife. If he starts growling or showing signs of bad behavior, get in his space again and back him up. Once he starts moving on his own, give him a command to sit/stay. Not sure how much he’s trained on those, but if he rolls over he’s likely been trained to sit/stay.
And as a previous member stated, stop having the wife throwing treats at him. Until you know his language well, you may very well be having your wife reward bad behaviors, continuing the cycle, and increasing his reactions to her.
15 months is still a puppy, GSD’s don’t mature until around 2-3yrs. You’ve got to mold the behaviors you want while he is still young enough to be moldable.
Invest in a plastic crate, or a wire metal one with a cover. Crate the dog before your wife gets home, and don’t let him out until he’s in a calm settled mood.
He is likely sensing your wife’s fear of him, and that puts him in a fight or flight mode. He is choosing the fight mode. Don’t give him the choice, make that choice for him. If he is reacting negatively to your wife, crate him or put him out back (if you have a fenced yard), and again, allow him out/in when the calm and settled behavior happens. I wouldn’t offer any treats to any of these scenarios. His treat is getting to be with you when he displays the proper behavior.
Once your wife is more comfortable, have her do the things above. Remind her he is a puppy in a grown body. His mind does not match his size. Sure, they look very scary, but what he is showing is lack of confidence (based on your descriptions).
And of course, a good trainer that has experience with the breed is always recommended. They can have eyes on the dog that we don’t have over a forum post. Just be careful in who you chose as a trainer. Find one that balances praise and correction. One extreme or the other can ruin the dog.
Most importantly, aside from the trainer, is to get it out of both yours and your wife’s head that he had a bad history with women, don’t go down that what if road. He could have just as easily been a great dog, and was surrendered due to death, poor health, a divorce, or military being shipped overseas.
You’ll likely get a ton of great advice here, wade through it all and pick what works for your situation. There isn’t a one correct way to handle or train in this situation. Good luck, and we all love pictures if you care to share!