I would put a selection of breakfast food on the table, tell her it's breakfast time then sit down and eat some yourself, not paying any attention to if she joins or not. I think 30 mins tops is long enough for breakfast. Give her a warning eg 5 more mins until breakfast time over and then put it away.
On the way to school is there anything she can eat eg cheese, fruit, nuts and raisins that is more of a snack time food that she will eat. Maybe what she has for snack at 11am. Do they have snack at school?
I think try to make a less of an issue as such for a few weeks, at least a month while she settles in. She won't starve.
Then if things are still not going well I would perhaps sit down and explain why it's important to have BF, back up with books. Then have a menu she can chose from eg scrambled eggs, cereal toast, smoothie anything she likes that she chooses before going to bed so you can have it ready in the morning. You could also use a sticker chart at this time if she eats the BF, or what I've done in the past is have hot chocolate for after eating.
My dd is only 2.4yo, but I have found that we have to adopt the approach that nannynz has suggested. We put her food down on the table, tell her it's there, and then make no other reference to it. Generally, she will start eating it. We can't even say, "Well done!" or anything when she is in the process of eating it, if we refer to it in any way, she will stop eating. On the mornings where she goes to nursery, I always get her up with plenty of time, and literally make her breakfast as soon as we get up, then leave it out for her. Cold toast? Yum
Dilemma, if it is a stroppy child issue not really a food issue as such, then you need to take the wind out of her sails by not reacting, hard though this may be.
Just try and ignore for a few weeks and see if the situation improves. I would give her a milky drink of some kind. Make really delicious breakfast foods like pancakes, jam on toast, fruit cocktail, whatever she likes, but then if she chooses not to eat, ignore. That is what I would do.
Some people just don't want breakfast though - my DH never eats breakfast, and can't stomach the thought of anything till 11.
I wouldn't go down the path of making anything special myself - you don't want to be stuck with making pancakes every morning.
Just offer your normal choices, and if she doesn't care for those, just leave it with no more comment than 'Ok'. It may just be about control - after all, she's found something that really presses your buttons. She won't starve through not having breakfast
My dc aren't morning people at all so they find it hard enough to get up and get ready. I manage to get ds to drink a glass of warm ovaltine and dd drinks a yoghurt drink which is OK, better than nothing. I'd like them to eat musli and toast so I always set it out on the table and sometimes they are tempted by it so that would be a good idea to start with.
If I just asked them what they wanted they would say nothing but if it's set out infront of them they sometimes eat.