Ooh this is a complicated one. Loads of children of her age decide they want to be vegetarian. As someone who was vegetarian myself for a while in my teens and twenties, and has a child (much older than your daughter and not living at home) who would be vegetarian but can't for medical reasons right now, I can understand this. And as a parent who has to cook family meals, I can see it from the other side too.
Firstly, I wouldn't let a 12 year old be deciding unilaterally what she would and would not eat. Unless she is prepared to take over the family budget and meal preparation, she needs to see that any decision she takes is going to have bigger implications for you than for her. Cooking balanced vegetarian meals is really rather hard - does she enjoy beans and pulses, tofu, etc? - because it can't be done just by dropping the meat from 'normal' meals.
Secondly, she is 12. She is going to need a whole lot of protein, iron and calcium over the next 4-5 years. Iron and protein, obviously; calcium because she needs to lay down bone mass in the 2 years before and the 2 years after starting her periods to try and offset osteoporosis later. The nutritional implications of becoming veggie at this age are probably greater for a girl than at any other she could choose! That is a really big deal for whoever is cooking for her.
So here is what we do here. We have always eaten a lot of vegetarian food and didn't want to stop that when we had children. So we often cook a hardcore vegetarian dish and when the children were little we would serve maybe some chicken pieces, cheese or even sausages with it, to make sure they got their iron and also because some very beany/pulsy dishes are very filling so they wouldn't really eat enough to get the nutrients they needed. Gradually as they got older they didn't need the additional protein any more so we phased it out. Now we probably eat two or three totally vegetarian meals a week and the rest are generally very veg heavy.
If you can spare the time, what I would suggest is trying to do one or two 'proper' vegetarian meals a week and see how you get on. I think it is essential that your daughter herself gets involved in the planning and preparation of these so she can see what is involved. It can be much harder work - I buy dried pulses as we don't like the taste of the tinned ones, and they need soaking and boiling, for example. Some of the delicious aubergine dishes need a lot of prep. But if she is going to be healthy, she needs to be able to embrace some of the more challenging ingredients and cooking techniques. She needs to learn about complete proteins and the difficulties of making sure she gets a complete range of the nutrients she would lose by dropping animal products. If you start slowly, and say OK look, we will meet you halfway but you have to help, and on the other days I expect you to eat what I cook, then maybe this will evolve naturally one way or the other.
The other thing I would say is that I too have massive issues with the standards of animal welfare and will only buy humanely farmed meat and eggs. We have a good butcher and a farm shop and while the costs can be marginally higher, if you use less of it and more veg, it goes a long way. Perhaps you can address some of her conscience concerns without consigning your entire family to a life of lentils and rice!
PS - had a wicked vegan chickpea curry last night, one of our favourites, so if you need any recipes...
