My daughter is only thirteen months & we have a Yorkie puppy who's only nine weeks. She's obviously not old enough to understand that she can't play rough with the puppy, so we never let them play together. I keep him in his play yard when she's crawling/walking around & that works very well for us. I still have lots of time to spend with my puppy so that he does not feel like he is unloved though
My daughter grew up around a three year old Chihuahua, so she is used to having a little dog around, & she also knows that she is NOT allowed to poke or tease my Chi. She gets in BIG trouble if she does this, but now she knows better. I know that my daughter is still very young, but there are certain things that she can understand that are not ok for her to do. We only tell her "no" to things very rarely, because we do not want her to feel that she can't do anything, so she knows that when we say "no" to something, it's a really big deal & she shouldn't do it. With that being said, I'm kind of disappointed in your vet saying that they can't get along! You just have to be EXTRA careful with your kids & the puppy around each other. Never leave them alone together. Don't allow them to carry or pick up the puppy. When they walk around, make them shuffle their feet so they won't accidentally step on the puppy. Tell them to look before they sit down to make sure that they don't sit on the puppy. Or if you don't feel that your kids are old enough to do this, get a play yard or an exercise pen & put everything in there that your puppy needs - a puppy pad, food, water, toys, a bed, a blanket, etc. I do believe that it is possible for your puppy & your kids to get along, so don't give up!
Amy