If you woke her say 7.20ish, would you have time to feed her before you leave for the school run? Then she'd be just about ready for a nap as you leave, so could nap in the car/pram? Will she transfer to cot or stay asleep in the carseat/pram when you get home? If you get a short nap because of the school run you could just make her next A time a bit shorter as pp said (say an hour rather than 1h15) and hope she does a bit longer nap so you can get back on track?
Weaning the 4/5am feed is totally up to you - the method the pp suggested is supposed to work well, though I just put up with that feed until DD dropped it herself.