My daugher (age 12) has been on the Ping since Nov '09, Type 1 since Dec '08. Her overnight BG started our good, then was high throughout Dec and Jan. We had adjustments given by her endo three times by phone, as we were still trying to find her optimal basal rates. The endo just increased her basal, the end of Jan at her quarterly appointment, and she has been much better. I can see from her experience how difficult your days are when you wake up high and try all day to bring your BG into a comfortable range.
I would suggest you give your endo a call, with a couple of weeks of BG readings. I would expect the doctor would be willing to work with you, giving you alternative evening/overnight basals to try, and having you test a couple of times during the night (I know how hard that is, but it will tell them what is happening overnight). They are the experts and we need them to guide us. They don’t have all the answers and it is your body, but they can help steer you in a better direction. A few other things we try: no carb foods after 8 pm, a higher I:C ratio at dinner; a higher basal in the late evening and into the 2 am time.
I would not try to adjust your basals on your own (as some people do), but it sounds to me like they are in need of an adjustment.
Everyone is different, and my daughter is still a child (growing, hormonal, etc). But think of it this way, when we sleep, our bodies repair and replace the cells we damage all day long. Our brains and bodies are being replenished. There is really a lot of activity happening, even though we look like we are at rest. Providing extra insulin overnight lets the BG enter the blood cells and gives the body the energy to perform all of these recovery functions.
I would really recommend you have your basals re-evaluated with your endo or CDE. It will likely help you sleep better, and awaken in a more reasonable range. It may take a couple of weeks, or even a couple of months. Insulin and BG are always in flux.
Good luck to you.