The naming of your DD is no one's business but yours and DH.
Please name your DD "Verity". It's a beautiful name and you will regret not choosing if you go with something else due to your mum's childish behaviour!
If your mum wants to go ahead and get a dog and call the dog "Verity", let her get on with it. It will make her look absolutely ridiculous and unreasonable when people ask why the dog has the same name as your DD and you go on to explain what your mother did.
If I were in your shoes, anytime my mother or sister made any kind of critical comment about the name, I would end the conversation, ignore them and refuse to engage. It I were on the phone, I'd hang up. If I were visiting them, I'd leave. If they were visiting me, I'd ask them to leave. They might then realise just how much it is upsetting you and how it is driving a wedge between you.
My DS is due in March. DP and I have really struggled to decide on names. We've come up with two.. Zachary and Jacob. Zachary is mine and DPs favourite. However, DPs mother does not like it at all and is pushing for Jacob. Who knows, we may end up changing our minds and going with Jacob anyway, but if we do, it will be our choice and not because we have allowed DPs mother to influence our decision. For now, we're settled on Zachary and no matter how much DPs mother screws her face up or shakes her head when she hears it.. tough! The name sticks unless DP and I decide otherwise.