At our house we use Almond Milk. It is a bit expensive but we find it to be worth it, and I use it for cooking and everything. So we most definitely drink it up.
Also, it sounds like your child gets a lot of congestion from the dairy (pretty common I think), so I am wondering if perhaps you could put a humidifier in her room??
It works wonders for DS and also for DH and me as well.
Oh, also, we went dairy free but kept yogurt in, because it's great as a probiotic, and it metabolizes a little differently than regular milk. So she might actually be able to tolerate it if you cut out the other dairy. Just a thought. It works for us.