Your have three questions in one now:
- Is a ketogenic diet appropriate for a one year old?
- Is a carbohydrate reduced diet appropriate for a one year old?
- Is milk really necessary for a one year old, or can I replace it with something else?
Milk is a very obvious nutritional choice for a small child. You mention that your child has been drinking formula so far, so the switch to cow milk makes sense. Milk is an excellent source of calcium, which kids need a lot of to promote growth. In many countries, it is also fortified with vitamin D, which enhances calcium absorption.
The Guidelines of the American family physicians recommends two or three servings (cups = approximately 600 milliliters in total) per day, to get around 700 milligrams of calcium. Except for fortified foods, dairy really is a very good source for calcium
Can it be replaced? Sure. There's children who are lactose tolerant or get eczema, for example, who can't have milk. Should you, with no physical reason to? I'd say no. For one thing, getting enough calcium without dairy really isn't all that easy, especially for a one year old. Two cups of milk are much easier to get into a toddler than 350 grams of broccoli.
Also, for the purpose of reducing carbohydrates, reducing milk intake is a weird choice. That 600 milliliters of milk has about 30 grams of lactose (the only carbohydrate in milk). That 350 grams of broccoli to replace it has about 25. Complex carbs instead of a disaccaride, but if you are concerned about carbohydrates, milk is just really not a bad choice anyway.
I don't think anyone would tell you to give your daughter pasta, rice, and nothing else. A child's diet should be balanced, so of course she can have lots of other things - vegetables, meat, etc. Without knowing what exactly you mean with a diet low in carbohydrates, it's hard to say for sure, though.
Someone else already linked to the study Progressive bone mineral content loss in children with intractable epilepsy treated with the ketogenic diet. The important part here is that this occurred even despite the children receiving calcium and vitamin D supplements and reducing their medication.
This study describes progressive loss of BMC in both the whole body and spine in children with IE treated with KD. These findings persisted after correction for both age and height. The decline in BMC occurred despite prescribed vitamin D and calcium supplementation and with a reduction in the number of AEDs used.
Most studies are of course done in kids with epilepsy - deviating from the recommended diet for children in such an extreme way is only done in studies where the benefits may outweigh the risks. Everything else is not defendable in front of an ethics committee.
However, in these studies, epileptic children on such a diet are compared to epileptic children on a normal diet. Differences between the groups are thus attributed to the diet.
Other problems associated with ketogenic diets in children are kidney stones.
Even in epileptic children, the ketogenic diet, despite its success, is only recommended after several medication options fail and includes regular checkups with a medical professional.
I think it helps to remember what ketogenesis is - fatty acid breakdown it order to get the body the energy it needs. That breakdown can lower the pH in the blood, leading to keto acidosis, which is dangerous. In a child, with a low body mass and a high energy need because of growth, this at least sounds very dangerous.
In summary, do not do this unless necessary. There is potentially no benefit at all, and a risk a parent should not be take on without medical need and supervision.