Hi E.,
I can tell you've put a lot of thought into this issue already. Here are a few points to ponder.
Your daughter is growing and changing all the time. She may be at a point developmentally where she feels challenged. She needs the space to learn how to deal with these big emotions when they come. You can help by having a gentle discussion when she is okay, helping her explore ways to recognize when a "melt down" is looming and different ways to vent these emotions. She might try punching a pillow, running up and down the stairs, dancing to a fast song, talking with a loved one, asking for a massage, etc.
Remember, too, that if you go down the path with her and allow yourself to show too much concern or big emotions, she will feed off that negative energy and feel even more upset or helpless.
She will learn in time how to recognize and prevent much of this if you have good care and input from whatever experts you see. Many kids have issues with emotions as they grow, so mastering them is part of maturing.
Since she is still just 6 years old, I'd also recommend trying to identify triggers for these episodes. You already know too little sleep plays a role, try to quantify how much is too little. Also, does this tend to happen when things are really noisy around the house or after watching a lot of telvision, for example? You might keep a brief journal noting circumstances around these episodes. The point is not to prevent them but perhaps help lessen their severity and frequency. Try a timer for a rest period, too. If she reads in bed until the timer rings, she is relaxing and resting even if she does not sleep.
If she is headed to school this year, be sure and work with her teacher to smooth things as much as possible.
Remember to bring hubby in on the discussion. Presenting a united front can help keep the two of you as a team and level things out for your daughter.
Remember also to check in with your pediatrician or other health expert if this continues for an extended period of time.
Good luck!
Parent Coach J. B