As a paleo diet follower who has worked at two different corporate daycare centers for a total of five years, I would highly suggest that you pack your child's own meals and snacks each day. I know this is a big hassle and is not fair given that you probably won't get a tuition discount for your extra efforts, but it is likely the only way to make sure that your child has a healthy, balanced diet and that you stay on good terms with the director and staff. You will probably need a doctor's note for this (especially if the nurse/director thinks she knows better), so hopefully you have a physician who would be willing to write a note.
If you tell the daycare workers that your child has some food intolerances and you think it would be easier to bring her meals/snacks from home, they likely won't be judgmental or irritated. If, however, you try to educate them on the paleo diet while requiring that they make huge dietary accommodations, they will likely think you're weird and annoying.
I have a couple reasons for this suggestion:
Their knowledge of food ingredients and food intolerances is likely to be EXTREMELY limited. My co-workers know I'm knowledgeable in this area, so they frequently ask me questions such as "It should be okay if I just scrape the cheese off the pizza, right?" in reference to the lunch of a child with a severe allergy to dairy and "What?! I didn't know croissants have wheat in them!" in reference to me politely declining the aforementioned croissants. Both of these comments, by the way, came from the cooks. They're not bad people. Some are even quite likeable, just clueless in this area.
A daycare employee's day can be extremely stressful, loud, fast-paced, and frustrating. In few places are the negative effects of population pressure more apparent than in a daycare. ;) I'm not saying other jobs aren't stressful or that this somehow excuses daycare workers from their mistakes, but I will say that I personally, even with my own dietary intolerances and deep understanding of these issues, have on SEVERAL occasions fed children foods they were allergic to. I work with toddlers (you didn't mention your child's age), so sometimes this means while I'm comforting a child who fell out of his chair and hit his head, behind my back a child who is not allowed to have dairy is grabbing milk from another child. Daycare directors/companies will swear up and down that they can make dietary accommodations, but this is not a guarantee. And the workers are frequently short on time and very distracted by all that is going on in the room.
That said, I'm sure there are daycares somewhere out there that would be able to accommodate your requests. Hopefully you have found one! I suspect they're in a very tiny minority. The two daycare centers I've worked at have been considered high-quality centers, and I still wouldn't trust them with a paleo diet. Good luck, hopefully others will have helpful suggestions! This is just how I would handle it with my future child given my experiences in the wonderful world of childcare.