Help with 3 Y.O’s preschool situation

My daughter has been going to preschool for a week now, she’s doing perfect regarding her separation from me currently, she understands that the school is only for kids and not for moms and that I will come and get her, school time is 9:30 am to 1:30 pm, she does well at first up until 12 pm hits and then she becomes sleepy and tired and cried at a teacher’s lab until she either sleeps on the teacher’s lab or I come and get her.
The school doesn’t have nap time, no other child need the nap except mine and the thing is she doesn’t even nap at home! She gets extremely tired and sleepy at around 4 pm which gets fixed with a little tv time, ofc she’ll sleep in a car at noon if the conditions help but I don’t encourage her naps because if she naps she’ll either have a very late bedtime or have a 2 hour wake up at 1 am or wake up at like 5 am.
The thing I put up together today was that I give her some screen time at 11 am too, as a wind down after playing and breakfast. So could it be her body’s used to this 11am quiet time that she feels the need to get away from children and stay alone at almost the same time? She’s an only child for now and we didn’t use to see other children daily, maybe 2/3 times a week.
I’m feeling frustrated, my thoughts are to suggest to the teacher some story time at 12 pm for a few minutes before they start their physical activity session at that time in hopes that my daughter gets the quiet time she needs, and tbh the preschool she’s going to rn isn’t the best, they have limited activities for kids and it’s what we could afford in the country we’re in. At least it’s only a course for a month and a half so it acts as tutorial for my toddler lol.
The teachers suggested I get her at 12 pm instead of 1:30 pm which feels kinda wrong to me, why would she need shorter time than others? Can the teachers do more or am I expecting more from them? Can stopping tv time at 11 am help her body adjust more? No other kid is making the behaviours my daughter is and I know we shouldn’t compare but I can’t help it sometimes, I’m also pregnant and fully emotional.