Oh God Claire, those 4.30 starts are hell. Okay so with Sam (yes, I remember it well) I would push him to 5 hours so a 9.30 nap uncapped (yes, I know) then he would sleep until 11 ish and I went for 5 pm BT. How about if you do the same this morning and if/when she short naps you try for another nap this afternoon. Or if she does a lovely long nap (hmm) you go for EBT.
HTH.x.