I have this too, I can take my BGL at 2am at 4am and it's on the lowish side but by the time I get up it's in the high teens. My DSN and Consultant have both said that the only way to deal with it is to go on an insulin pump as you can adjust the basal for the times you need to. So I have now been assessed and am on the waiting list - probably around another 6 months to go. In the meantime I have been taking my breakfast QA and Basal before I get out of bed and this means I don't go quite so high, also if I can face it I set my alarm for sometime in the night (have been doing 4am but I'm usually still have a BGL of around 4 then so need to try 5am).
I've also tried taking night time basal later - and taking humulin S at dinner as it lasts 6 hours instead of 4 - didn't like this too much though as I kept having night time hypos so now back to humalog.
Think I have to agree with my DSN/consultant - the only effective way must be a pump - can you try and get one?