VQ I have no useful advice other than to keep speaking to school and being insistent with the issue - but I just wanted to say it sounds awful for your daughter. I have been so lucky and never experienced any issues in school ever, and wouldn't know what to do if I did.
Hats - take one. Even though the wards are hot, some babies have difficulty in adapting to the temperature, and regulating their own. Your baby is much more likely to get too cold than too hot. If a baby gets cold, it can affect feeding. Babies who are cold will not be receptive to feeding, because their bodies are trying to stay warm, feeding becomes unimportant. For those wanting to breastfeed, this is especially essential to keep them warm, because its hard to get a cold baby to wake for a feed. With a bottle you can make them feed, but they have to work for a breast feed!
I don't routinely put a hat on a newborn unless its prem, or low birth weight. But I might advise a mum to put one on if the baby becomes cold, or is otherwise at risk of low temp (prolonged rupture of membranes, meconium stained liquor, gbs+ mum). After the first 24 hours most babies are regulating their own temp. :) So if I was going in, I'd be packing 2 hats, a few blankets of my own, 2 cardigans and at least 8 vests plus babygrows, because its amazing how many they grow through, and in the middle of the night, you don't want to have to wait for someone to bring them in! I'd pack for a 2 night stay. Just in case.
I will probably have a vest/babygrow/cardigan/hat on this baby when it is born, plus wrapped in a couple of blankets. But that is in my own home, and it probably won't be as warm as hospital!
I find the babies that tend to drop temps are the very mucousy ones that end up with 'wet necks' which sits on the babygrow/vest. A lot of people just leave them, but I'd always change, wet clothing will not keep them warm!
So, no, you don't need a hat for a full term healthy baby, but you don't know what might happen, so take one! The midwife can advise you at the time. I do get asked, and sometimes I say yes, sometimes no, it isn't needed! And yes, I'd use a hat/hood for the trip home, especially on a cooler day.
Oh, and a lot of the hats are too big, so take a small size as well as a bigger one 
Wow, that became an essay 