Absolutely rehome the dog, the dog will be so much happier in a place where it's better care for. Dogs like to know the rules and know that they're being good, that makes them really happy. But if they don't know the rules due to lack of training or enforcement, then they don't know how to be good, and then instead of feeling happy and secure, they feel confused and nervous. And it's not good for anyone in your house to be living in such a tense situation all day every day.

Your son won't see it right away, but it will be so much better for everyone if you rehome the dog. I would start by contacting the people you got him from, they would either take him back from you or help you figure out what other option would be best for rehoming. They would rather help you do this properly than have you do something like giving the dog away on Craigslist for free, and they want what's best for doggo too.

I'm sure the fighting with your son won't be resolved overnight once the dog is gone, from what you've written it sounds like he's going to be mad. But in other ways, the situation will improve right away. You won't have to clean up after the dog or do anything to care for it, and the dog will be on its way to a happier situation. I wish you well, OP. You sound like you are stretched pretty thin and could use something in your life getting easier. This is the right thing to do any way you look at it.