Op it sounds to me like your pre-school really don't want to help. Yes, it's unsettling for the adults to have to do something differently and not follow a routine that's comfortable for them but a child's medical needs need to come first. It may mean a change in routine for the whole setting for the sessions your dd is. Sometimes things are going to need to be done in a different way so that all children can be included instead of excluded.
The staff aren't inspiring your confidence in their ability to properly care for your dd's needs so it's not at all surprising you're feeling anxious and needing to check on her while she's in their care. It will disrupt the session but you need to know you can trust them to monitor dd's condition correctly and react as needed. They have been trained and are responsible for a little girl who could become very unwell very quickly, so should understand what needs to be done without checking with you.
If they wanted to they could easily accommodated dd's needs without it being detrimental to the other children's development and care but it would mean change for them on a big scale. Currently it sounds like they are trying to fit dd into what they have always done, not ensure she is able to access their core provision and have her needs met. She is becoming a problem to them because they are making her the problem.
I started work in a nursery that had done things the same way for 14 years. The routine was pinned on the wall and everything about it was done the same way as when it first opened. Children's snacks, lunches, free play, creative, physical, sleep ect was all approached in the same way. When new children started with us they were conditioned to fit in with the nursery routine. Because of this the staff who had done things the same way all their working life and liked having lunch at the exact same time, getting their coffee at x point past each hour and doing things the way they always had. It worked fine until a child was enrolled in the nursery who couldn't function within the established routine. Things had to be done difirently for that child to be able to access the provision.
Sometimes it isn't just a case of changing what is done slightly. Things might need overhauling completely. This is what had to happen in the nursery I was in. Snack time had to change and run for 90% of the session. Time had to be made to do medication or testing at a set time. Staff had to take breaks and lunch on a rota, not all together. Children's lunches were approached diffrently. Extra trips were made to the toilet. The children were still doing the activities and using the equipment as they had always done but what the staff did was different. All the children were happy, learning and well cared for as they'd always been. But the adults had to adapt and for lots of them it was uncomfortable.
So yes the staff at the pre-school have other children to look after as well as dd. But with a little more flexabilty it can all be done. If I was looking after your dd I'd personally be greatful for the chance to get the full handover from you without feeling like I needed to rush you to talk with other parents, I would just need to make sure I was able to manage my time and work load to be free at your regular drop off time.
Personally I would encourage you to go straight to the pre school as others have and do what you need to do there. But I'd expect you to be able to do it in the pre school room so your dd has time to settle with you there and you can do blood sugar and eat together. It would probably be very inconvient for the preschool and make them encourage a late drop off or magically make things easier for you. For some children it might ease the separation angsity and make them keen to attend so more willing to eat breakfast/help you leave on time. But only you know what dd is like and what is best for you in your personal situation.
Perhaps ask the manager to surgest things the pre school could do to help you, for example allowing you in early to sort out dd's snacks and blood sugar so she's ready to go as/when other children arrive. Also if you have the strength or time to do so think about what they could do to help you feel less anxious and build trust. Would it help if the manager made time to phone you each morning for a few weeks to let you know that the blood sugar reading has been done. Can you write down circumstances in which you'd like to be phoned if dd has a problem. Or if staff could reflect their understanding back to you in conversation; so if she has low blood sugar I'll make sure she has some juice and someone phones you. Could they create a home school book for your morning handover notes. Talk you through their plans to manage dd's condition. If you can show willingness to trail things for a short period to see if they work for you.
You must feel in some ways this is more trouble than it's worth. But your dd is enjoying activities that she doesn't get the chance to do at home. With the support of the pre school it has the potential to be a really positive place and experience for her. Even without her medical problems it would be perfectly normal for her to get upset when you leave. Helping a child deal with that is part of working in the early years. I do wonder if it would be helpful for dd to have a real clear routine in the pre school because if she could be finding it stressful if testing her sugar levels isn't done in a consistent way.
When I looked after a child her age with diabetes the nurse who trained us all came in a helped to test the blood sugar levels of all the children in the room. It made the management of that child a none event afterwards. Things like short stories were planed around when we needed to do sugar levels in such a way the child could still be involved and given snacks ect as need, sometimes as much as 8 or 9 times on a long day.