How can I convince my mom to get rid of her dog?

You are both at fault here. Peanut senses your disdain towards him and acts out. But the last thing that dog needs is to go to another strange home, or worse: a shelter. What he really needs is a good trainer. There is a root to all the aggression problems. He may be scared, so he is attacking first so he won't be attacked (dogs attack fear, so your dogs would have most likely attacked him). You may not want to see it, but it is most often other dogs that provoke aggression from other dogs. If at any time you see the dogs staring at each other, you must intervene and snap them out of it. Try to help the dog, not prove to your mother that you are right.