I agree with Ghoulsareloud.
Hitting a dog that is growling at children is the worst thing you can do. Obviously your qualification is in animal behaviour and not canine.... Dog isn't comfortable around children, dog gets hit around children, so dog starts to think 'well, children really are awful,. I get hit when they are around.
It's not rocket science. Dog growls at child. That is the dogs way of saying, 'I am not happy in this situation'. At that point it would have been sensible to remove that situation, by shutting the dog in another room if needed, or keeping the dog on a lead and keeping child away. However, instead you chose to still allow your child near to the dog, it carried on growling, no-one listened or paid attention to the fact the dog wasn't happy, so the dog did the only thing it could and progressed to the next level, snapping. It was a warning, if the dog had meant harm, it would have done harm. It is not the fault of the dog, but the fault of the adults around it that allowed it to progress as it did.
I don't think you are being especially unreasonable by asking MIl not to bring the dog to the house, but it is compeltely unreasonable, and ridiculous to hit it. The fact the dog didn't retaliate shows the dog is actually pretty good natured.
IMO, the best cours eo faction would be to socialise the dog with your children in a positive and safe way, so starting with the dog behind a stairgate, children standing nearby (not too near, so the dog is comfortable) and an adult giving the dog treats. Gradually allow the children nearer, and eventually witht he children giving the dog treats through the stairgate. I would suggest contacting a good behaviourist via the APDT website in order for advice on this though.
Of course, theo ther solution is to keep the dog away from children, but your MIL still needs to address the problem. A dog that is scared or anxious around children (which is what this dog sounds like) is dangerous, not least, because situations like this happen, where adults ignore the warning signs.