I have a dcb who has been with me since November. He is also a poor napper and his naps really range in length. I let him cry when he wakes up early. I have let him cry for up to an hour when he wakes early and about half the time he does go back to sleep. Mom knows that I let him cry, but not how much. He is not on a consistent schedule at home and has a lot of sleep props at home. At my house I follow a schedule and don't allow props so in essence he was sleep trained here. His sleep has improved a lot and sounds like it has improved at home.
I would not give up with this child. At her age it seems that one hour of sleep all day is definitely not enough. It may be that 12 is too late for her nap and she is overtired. Is it possible to put her down at 11 after a big morning snack? Not sure when your snack time is. It might be that she needs an earlier nap time if she is transitioning from 2 to 1 nap. Maybe after a few weeks of earlier nap you can slowly move nap time back to 12. Sometimes I have to rejig my schedule temporarily just to deal with nap transitions. Usually the older kids can go to bed a bit earlier and they still fall asleep. Then we have a big afternoon snack and though my break is a bit earlier than I prefer it's better than not getting one!
Good luck, it is very frustrating to deal with short nappers!