What worries me here (trainer/behaviourist hat on, so this is my main job!)..
You say your 3 year old is obsessed with animals - unfortunately when you bring a puppy into the house the ideal situation is that your small child and the puppy spend v little time together that is not 100% supervised and that both are reinforced heavily for ignoring one another as much as possible.
Kids who are obsessed with dogs are what we call 'magnetised' to them, they want to approach, they want to touch, they want to grab and hug and stroke and poke and and and...
All things that are going to go badly with any dog but particularly a puppy who learns about the world by biting it, and everything in it - and you cannot STOP him doing that, he needs to do it to learn how to use his mouth, how not to use it, etc etc.
Adding a puppy now, is going to be like adding another child, so you have two, who cannot communicate the same language, cannot communiate fluently or effectively with YOU either, get frustrated because of this, will make mistakes/have accidents because of this (hitting, biting etc, both of them!).
And then on top of that the third party is a child who is only there part time and is currently scared of dogs and her other parent isn't supporting fixing that.
Unless you have the ability to be in three places at once, genuine eyes in the back of your head, and some serious training skills... I'd go with no, wait, another couple of years.
I would however, go to LOTS of animal based places, do lots of stories and talk about animals and how they need space and calm and respect and so on, with BOTH children.
In the run up to actually looking for a puppy, or older dog, I would locate good training classes near you (force free, positive reinforcement based, I strongly recommend the Puppy School franchises, they are VERY good) and ask if you can come along and sit in a corner and simply watch puppy classes, with BOTH children.
A good and sympathetic trainer should be happy (I would offer them some payment for their time of course) to help you get both kids used to dogs and the fearful one some way toward addressing her fears.
Then you could progress to going for walks with suitable friends dogs perhaps, so that they get used to dogs being around WITHOUT the intense pressure of it being their own puppy that they want to cuddle and play with alllllllllll the time.
Raising a puppy correctly is hard work, far harder than many people realise, if you want it to be enjoyable and fun for everyone, I would wait.