I would train the puppy to be gentle.
Train GENTLE with treats.
Start by saying GENTLE and then offering a treat with your open palm -- treat in the middle. The pup has to take it gently because they just don't take a wide open hand, they put their mouth on the treat and take it.
Say "Good Gentle." and repeat this, several times a day, for 4-7 days. Always use the word GENTLE before, and Good Gentle afterwards.
There is no hurry. You are teaching the pup to associate a word with an action of being careful. Take your time.
When he is good at this, make it a little harder. Put your thumb over the treat. Say GENTLE, and offer it with the thumb over it. If he bites your thumb, say, Eh, No! and then try again, GENTLE in a soft but firm tone. When he takes it Gently, Good Gentle.
Repeat, repeat, many times 4-7 days. No hurry.
When this is 100%, move on and close your fist. Say GENTLE and offer your closed fist. He is smart and has a good nose, he knows there is a treat there. If he bites your fist, "Eh, no!" and put it in your pocket. Try again in half an hour or so. If he licks your hand, open it and say, "Good Gentle."
Repeat, repeat, repeat until he is 100% and so that you have don this many times over 4-7 days. Continue to remind him, continue to say good gentle every single time.
Now, make it tougher, put it inbetween your thumb and fore finger. Tell him, GENTLE, and watch him delicately take it out of your fingers. Good Gentle. When he is perfect at this point, the command -- the word, should be associated to being careful with his teeth. Now, you can refer to other things with the Gentle command. Gentle with my fingers. Gentle with the baby.
And if he is pushy, remind him, GENTLE. If he continues to be pushy, do not give him a treat. In fact, training or no training do not treat the dog if he is being too pushy about getting it. You are rewarding his pushiness and creating a monster.
Teach him to always take treats gently. He'll get it. He's smart.
Also, you have a dog now that seems to be too treat motivated. I know he is a baby, and that shouldn't be a consideration at this stage. I agree with substituting for a toy, but do give treats to lure him into position for new tasks, and for doing something exceptionally well. It is called phasing treats out.
If you know what clicker training is, they communicate that the dog did the right thing by clicking with the clicker. Well, the dog wouldn't give a patootie about the clicker if they did not "load" the clicker. They do this by clicking and giving a treat. Well, I don't want to carry clickers around and have to have my hand on some gadget to communicate with my dog. In the early part of training, I use treats to load my praise. Works anyway with praise, treats or no, because you can make your voice happy. But treats are even more rewarding at some points. So when you treat your dog, you tell him Good boy, what a good dog you are, good Gentle, good boy. And this is all in the tones that that extraordinary instrument you were born with can make that lets the dog know he did a good thing.
So we load our voice, our praise with rewards. And then we start phasing out the treats. We start giving treats every time, and then we back down to every other time, or every third time. Or, if obedience is our thing, we start giving treats for the best sit, the straightest sit, the quickest down, the perfect come-front. Yours is a baby yet, so all of that will come in time.
But the baby is giving you an indication that treats might become a problem for him. So you stop giving them when he pushes. You say, "Eh! No!" You have to have your timing right. Say Good boy, immediately when he completes the task satisfactorily, and reach in your pocket, and say "Wait" get it out and say GENTLE, and offer it to the dog. He will learn that he cannot push you to get a treat.
Try not to let treats get to be a big deal. And use your voice to let him know the measure of his success. Praise can be a quick, good boy, or a total party, "Good boy, Fido! what a good boy you are! You are such a good boy. Let's do that again!" and everywhere in between. It can be a low toned, good boy to let him know he did the right thing, ordinary, what we expect. Or high pitched and excited to amp him up a bit or to let him know he did really good. Your voice is a powerful tool.
If you cannot speak, I am sorry. But you can get him excited teaching him totally in sign language as well, using pets, and signs to let him know he did good too. And Smiles -- they look at our face.
Another game with the treats to do, LOOK or WATCH Me. Put treats in both fists, and the pup will go for one or the other. when he looks at your face, offer him a treat and say, Good Look. What you are doing is teaching him to get his cues from your face. And to focus on you. Dogs know the difference between frowny faces, and Happy faces. They are quite intelligent.
Bear Cub, Hepsi-Pepsi
Cujo2, Karma Chameleon
Ramona the Pest, Kojak -- who loves you baby?
Tiny Tinnie, Susie's Uzzi, Kaiah -- The Baby Monster.