Oh dear. That is so sad. I'm so sorry what a dreadful birthday.
Even my piece of doo doo never stooped that low.
To me you are at a pinch point in your relationship. I would suggest you see relate on your own to talk over your feelings and to see where you want to go with this relationship. After the first cunt's been let out of the bag, I think the second and third one will slip out much easier...eventually as more and more filters come off ...possibly/probably in front of your daughter.
If you feel there is enough in the relationship to salvage then you could go together as a couple and work on the flashpoints .....that is if he is really committed to having a nicer pleasanter relationship.
What I don't think you can do is sweep this one under the carpet. It will only get worse.
It's emotional abuse and no way should you set the precedent for your daughter.
Today is the first day of your new life , and a new year for you, so make it a good one.. Get a quiet space. Get some thinking time. Write a list of what you want then look at ways at how you can achieve your goals. You might be a couple but you are still an individual in there and you deserve to be valued, respected, cherished, loved ....if you arent getting these things then you need to make plans.
Somehow, I think this birthday is going to count. Happy birthday OP.