Part of the problem from the pre schools POV could be that your dd is missing out on circle time or whatever by being there late.
They might be in a position where they genuinely can't be flexible without it being detrimental to the whole group, and that goes beyond reasonable adjustment.
In the pre school I used to work in we very much did free flow, but there was still some kind of structure to the session. A child arriving late would have missed the bit where we talk about the date and do a register and that kind of thing. It was only five minutes sitting, and children weren't forced to do it, but if they were never there then they would never have the opportunity to participate in that part of the day, which was one of the things that really does make a difference to children settling at school.
A distressed child who then arrives after that is not only missing out on a learning opportunity, they are also taking a member of staff away from other children who are engaged in an activity and who need adult support to get the most from it.
In my experience, it's really not as simple as just 'making an exception' because making that exception has detrimental consequences both to the child that is late and the others.
I have seen plenty of children become unsettled because of another child being distressed, and because of parents turning up in the middle of the session. It does happen, and it's really not fair on those very small children to dismiss it as if it is unimportant.
I think we need more information from the pre school about exactly why they find it disruptive. They may well have a valid point, but depending on how they structure their sessions, they may not.
It does sound like they are not being helpful in ways that they could be. The milk thing is one of those issues that there are ways round if they want to be creative. So is the issue with the biscuits. They should be making more effort in those departments. In this case it sounds like both the nursery and the parent need to compromise, for the benefit of all the children, but that means give and take on both sides, not just a blanket insistence that the pre school staff have to do what the parent wants.
Maybe agreeing a time, albeit a late one would work. And if OPs dd hasn't arrived by a given time, then they can know not to expect her. It could be that the most difficult thing for the pre school is that they don't know what time OP is going to arrive each day which is making it impossible for them to plan properly.
Maybe going to pre school on the flexible day and choosing two other mornings would work better. Many children don't do every day until a term before they start school anyway.