It seems to me like she's most of the way to one nap but not quite ready to do it every day. Could you maybe switch between one and two nap days depending what time she wakes up? I know it's a bit of a faff and you might prefer going cold turkey to one nap, but if that made things worse I would try to go with the flow a bit more. I did it for about 3 months with my DD1 so if she woke before 6:30 we would do 2 naps and otherwise one. After that she was at the point where she could make it to lunchtime either way.
So with yours, say if she is up before 5:30 you do give her 20mins or so in the car on the way back from school, and then put her down for her 'real' nap about 2.5hrs later. Would she have time for a good 1.5hrs if she sleeps that long? This is why I keep asking about school run times, I'm still not sure when you have to be out! But it could look something like this:
WU 5:30
Nap 10-10:20
Nap 1-2:30/3
BT 7
If she wakes up later in the morning you know she's ok to make it to 12:00 as she was doing before, and presumably you can let her sleep for as long as she wants then?