Oh I am so sorry for the rough time you are having - hugs! I agree with you 100% that she needs the sleep - you know best - and from what you say, when she does sleep, it is great and she's happy. Same with my son. I hope the doctors can help with something.
We just came out of a really difficult spell with our son, now 2 years old. We finally fixed the night wakings and early wakings but not sure if it would work for you. However, I will at least tell you what helped us, and maybe something can work for you or give you new ideas.
What helped us:
1. 5 pm dinner, 6 pm bath, 630 pm yogurt, 7 pm bedtime
2. nap from 130 - 3pm
3. plenty of food during the day
4. reward stickers (for new trains) for good sleep
5. hour before bed is quiet, low-key
6. we don't go in his room in the middle of the night anymore if he's fussing/crying - we let him settle himself now, and it is the BEST thing we ever did. We used pu/pd, controlled crying for almost 2 years and it just was not working anymore. He was expecting us to help him get back to sleep ALL the time, even if it was just a 1 minute visit to fix a blanket. Now that he's over his reflux and it's just habit, we had to break that habit. (Oddly enough, fixing the night wakings FIXED the early wakings, so maybe one might help the other for you??)
7. Same routine every day so he knows what to expect. If his days are always unpredictable, he sleeps horribly and is cranky and tired.
My son always wakes from naps a little tired a cranky, sometimes ALOT tired and very cranky - for about 30-60 minutes at times! I just think it's a phase.
I hope something changes SOON for you - I know how you feel. HUGS to you!!!