I don't think you should cut the nap either. What time is the nap? Waking up early can be from overtiredness, although it is very common for toddlers to go through 5am wake-ups. Generally you don't want her to be awake for more than 5.5hrs of wake up time. Does she act tired during the day?
What do you do when is up that early? I would do wi/wo until the time you want her to wake up. Not sure if you feed her as soon as she is up, but if you do, I would cut that out as well as it can set their tummies on "breakfast" mode at 5am.
Does she sleep well at night? I wouldn't push the bedtime. That can and usually backfires and she will wake up even earlier and will have restless sleep at night.
Layla