YAB a bit U, but I can hear that you are genuinely concerned here and only wanting the best for your daughter. Your DP is being very unreasonable.
First of all, I think you should be proud that she is trying to make ethical choices and to be a positive change in the world around her. That's an amazing quality, and she deserves praise. Sometimes I look at young people nowadays and they blow me away with their commitment and activity. Your DP needs to recognise this. I would also be a bit
about his protestations - are you sure he's not just trying to avoid eating more vegan food himself?
Secondly, there is no reason to think that she will end up with health problems because of a vegan diet, PROVIDED that she is careful. I would treat this as a great opportunity to teach her about nutrition (and perhaps educate yourself as well - I certainly don't know everything you'd need to know to run a vegan diet!). That way, you'll be on board with her choices, and you can ensure that she gets sources of the nutrients that she might lack, and it might also benefit the rest of the family too. (I know from my vegan friends that you can make a kind of nut and seed mix, where you take a spoon a day to get oils).
I would also address the cost issue by giving her a small slice of the food budget each week and encouraging her to get involved with meal planning, cooking, and purchasing of food. She could maybe cook for everyone once a week on a Sunday or something. It'll be good training for when she leave home, and will give her new skills, as well as making her understand that you have to do the legwork to be ethical - you can't just make choices and expect other people to pick up the work that's involved.
Also, this needn't be a massive hassle. Most vegan meals are super-healthy - think stews like caponata, curries, bakes - and will be something the entire family can actually enjoy a couple of times a week. If you cook for an extra couple of people and freeze up the left-overs, then you will have a supply of small one-person meals for your DD which will save cooking a separate meal for her each day.