No, you should not rehome the dog. It is both your faults the dog poops in a cage. A well trained dog does not need to be crated. I hate crate training. No one potty trained the dog. Why did you get a dog & then not train it? The dog needs to get out of that damned cage. In my eyes you could put the baby in the crate & leave it there for hours on end. Ignore it.
Why don't people train their dogs? It is mostly for our sake dogs are trained. I want & demand a dog to be 100% obedient. I am a hard task master & I demand.
Three main things you teach a dog. When you first get it you potty train it to go potty outside & you teach it not to bite. Then Basic Obedience, 7 commands that will teach your dog to be obedient.
Don't know if you didn't know to train a dog or that dogs were trainable. Is he just a dog with no training? He can be so so so much more if trained.
How is a baby going to get into dog poop? You have plenty of time to train the dog before the baby is crawling around. That dog was your baby first & now you want to get rid of it. It don't make sense. Join an obedience class & in 8 week you have an obedience trained dog. But you may need to be a hard task master. Obedience Training is obligatory, the dog learns, period.
All of our learning is repetition, doing something over & over & over until you 'perfect' it. Training an animal is the same thing. Our children's schooling is all repetition.
That baby is going to be his baby too. They should meet the moment you come in the door with the baby. Get down to the dogs level & let him sniff, kiss, inspect the new member of the family. Some day that dog may be your best baby sitter. You will have to teach them both to be gentle with each other. & don't forget you have a dog that wants to love that baby too.