Most commercial vitamins don't have nearly enough in them to make much difference. They carry warning labels too, because they aren't balanced enough to be metabolized as food, so you can get way too much of a particular ingredient. They might make you feel better to give them, but they aren't doing much. Because they aren't absorbable, many pediatricians and most food scientists consider them to be "expensive urine" in that much of them are just eliminated. Fifty years ago we could get our full nutrition through our food - and there are still people who cling to that ("Eat healthy foods and you'll be fine") but even the american Medical Association disagrees with that. They came out with that finding over a dozen years ago. So depending on what a pediatrician has read and keeps up with, you'll get different opinions on whether to take them or not.
It's absolutely not solid science to say that a blood test will tell you what you are missing and then you just take that one vitamin or mineral to balance out. That's not the way cellular biology works, and it's not the way the cells absorb nutrients.
I would go much more toward a comprehensive formula with multiple ingredients (the healthy human cell needs about 72 vitamins, minerals, trace elements, phytonutrients, etc.) for proper functioning. If it's balanced, it is metabolized as food and there is no possibility of overdose as with any supplement that is required to have a warning label ("do not exceed recommended dosage" or "keep out of reach of children"). This is what we do with preemies, with kids who have Failure To Thrive syndrome, and kids like your daughter. We also do it with pregnant women who are nauseated/vomiting and absorbing very little, and with cancer patients who are ill from chemo and radiation. And many more. It's delicious anyway, and can also be added to other things like yogurt and smoothies.
I also did a lot of "hidden nutrition" with my son, who had issues with texture. You'd be amazed what you can do with a lot of easily available foods that kids don't think they like but which can easily be combined with others foods. My kid was eating tofu and broccoli in ways you would never imagine. I agree with you about the milk, by the way - it's filling her up and not giving her enough to make up for that.