I've read contradicting claims about whether you can give skim milk for toddlers over 2, or just low-fat. In fact, my daughter is two and a half and only 12 kg, so our goal would be to gain weight, and she gets a better appetite after skim milk. Should we give low-fat or skim milk?
Dr David Ludwig and Dr Walter Willet published a 2013 opinion column in JAMA Pediatrics that questioned the value of recommending low-fat or skim milk for children. While much of the article is behind a paywall, I did find a discussion of the opinion column (and it was also discussed on Time).
Ludwig and Willett argued in their paper that children who don’t feel satiated from a glass of skim milk—which contains 83 calories and no fat—may be inclined to eat an extra cookie or two as compared to when they drink whole milk, with its 8 grams of fat and 150 calories.
“We just don’t see any benefit for focusing on reducing fat,” Ludwig said. “We think it’s a holdover from a paradigm that evolved in the late 20th century based on the relatively simplistic idea that fat has the most calories per gram and that eliminating fat will reduce weight gain.”
The explanation for why skim milk is correlated to higher BMI appears to be due to the fat content. The saturated fat found in whole milk contributes to a feeling of fullness; kids who drink skim milk don't get satiated as quickly, and get hungry again more quickly.
You're coming at it from a slightly different perspective from many parents: rather than thinking about cutting down on milk fat to reduce obesity (which evidence suggests doesn't work), you seem to be thinking about using instead encouraging more calorie intake (which, from both your experience and what research seems to indicate, might work). However, from a glance at developmental charts, your daughter is about 25th percentile. That's below average, but not (as far as I can tell) underweight or failing to thrive.
If you're finding that she eats less food with whole or 2% milk because she feels full more quickly, you could also consider switching things up: serve milk partway through the meal, or as a snack between mealtimes. This is more an area of personal preference than science, though. There are fat-soluble vitamins naturally found in milk fat which need to be added back in when the fat is removed, and I personally tend towards a less-processed choice.
This could probably be a question from your pediatrician but I could say that we gave our kids whole milk until they were about 2 then switched them to 2%. Our boys have no problem gaining weight though. In fact they are a little on the chubby side. If weight gain is your goal then I would stay with whole milk for a little longer.