Firstly OP, congratulations, exciting and fun times for you all.
Secondly (and more importantly) what a cutie the princess is 🐈⬛
But down to business, when introducing 2 pets it's important to keep things as stress free as possible. I know that's not always easy but remember that if the cat or dog senses stress from you (or your partner) then they will think that something is wrong and will be in a heightened state, so keep things chilled.
Preparation is key and as your cat is a house cat (you probably want to do the meeting in your house) I would suggest giving your cat a cuddle or something to make sure she is awake and aware of her surroundings before your partner's dog first comes in. All the doors to the other rooms should be open so that your cat can escape if she feels threatened. The dog should be on a lead. He will be excited because most dogs are excited to explore new spaces. Try to calm him down.
When the dog comes in, stay calm about what is going on. If they stay in the same vicinity reward each of them, treats, toys, affection (whatever floats their boat) while keeping things calm however, it's more likely that your cat will run for cover and your partner's dog will bark and want to play but dont worry, settle the dog down again and give the cat time.
This will take quite a few go's so dont be disheartened and take your time. As I said earlier, dont be stressed. If things start getting stressed, stop what you're doing, take a break for everyone to calm down and try again.
These interactions should be a short and quick and gets repeated often.....hopefully won't take long to start seeing improvements and longer interactions between them.
If things get heated, do not pick up your cat but deflect the dog with distraction, your cat will be better able to remove herself from a situation than you.
I have given my cats a cupboard with a cat flap as their safe space, which means my dog can't get in...maybe think about something like that too, ie a safe space for her? Remember, its not a bad thing if your cat decides she never wants to integrate with the dog, some cats just like their own space 😎
I am not a trainer but I used to have a Pet shop (in Andover, Hampshire a few years ago) and I would host training nights where a trainer would come and help customers out, I've had and have lots of pets which has taken some integration (dogs, cats, chicken and ducks all needing integrating). I would also recommend speaking to the dogs trust as they have trainers and sometimes have training classes too where you can get some help.
I hope you get a peaceful but fun home as loving pets make the home