Not only is it concerning that your wife is physical with the dog in your presence, but I also wonder what happens when you aren't home. If she has no problems "training" the dog in that manner right in front of you, despite your objections, I can't imagine how much worse it is if she's left alone with it.

Either way, it is your job to protect your dog. You took on the responsibility and you are the caretaker. So, I think you need to ask yourself "am I in a state where I can responsibly give this dog a safe home?". If your wife is beating the dog and you can't stop her, then the answer is a resounding NO. Personally, I would consider rehoming the dog. It sounds as if you need to concentrate on getting other aspects of your life together before you can be a pet owner. You're obviously not happy, so what's stopping you from divorcing and moving on? If my spouse beat one of my animals, they would have been out the door within the blink of an eye. The fact that you're staying with her despite all of the glaring red flags leads me to conclude that you have some personal exploration to do in context to your self esteem and your happiness.

Find a home for the dog where it will not be physically abused. To keep it at this point would just be selfish, as your wife is obviously not going to stop abusing the dog. Unless you plan on leaving in the immediate future (or kicking her out), your only option is to put the dog's safety FIRST and find it a better home. And I don't mean give it away to the first person who wants a free dog online, because that's a dangerous scenario. I mean you need to actually do some leg work. Call local foster agencies, etc where they can take the dog until adoption is facilitated. Adoption agencies like these always do background checks to ensure the animals go to a good home. Your dog deserves better. Good luck with everything.