YANBU. I would not have paid for the bear. I find some of these replies baffling- yes toddlers have tantrums, but you deal with them appropriately rather than excusing the behaviour.
The OP has clearly said her dog is not aggressive or that he dislikes people. She has specifically said the dog is quite sociable and likes people, but is quite bouncy and she didn't want him to knock over the little boy. Therefore, when asked if he could pat the dog, OP said she'd rather he didn't. That is being a responsible dog owner. And if the cafe are happy for OP and her dog to be there, then she is free to take her dog there.
Mum's response was to wind up her toddler by telling her the dog was nasty and then she was rude to the OP. Toddler's response to being told no was to have a tantrum and then to try and kick the dog. OP then had to make the child go back to his mum- if mum was so worried the dog was "nasty" then surely she would have made sure her child was nowhere near it?
Yes, toddlers have tantrums and that is normal. That doesn't mean that it doesn't need to be tackled. Not only that, but I don't buy that "a 3 year old doesn't know any better"- my DD knew from a young age that kicking/hitting or otherwise mistreating animals was not on. To my knowledge, she never has done so (and if she had, there would have been some sort of consequences to her actions). This is not only because I think it is important children learn respect for animals, but also because it is potentially dangerous for her to be aggressive towards animals as they may react badly. It's not rocket science!
Dogs play with toys, they will chew and rip them. If someone threw a toy at our dog he would think it was being given to him and he'd chew it. He is definitely not a vicious or dangerous for a dog and is very well trained. The little boy threw the bear at the dog, and I agree that he is unlikely to have understood the implication of throwing the bear at the dog, but even a 3 year old has to learn that actions have consequences. Mum could have explained to her DS that he shouldn't throw things at other people (or pets), and if he does then these things may get damaged or lost.