I'm on DAFNE right now, and that's a classic scenario to manage. Do a test at 3am, because that is when the hypo traditionally hits (you said it was 6 at midnight, I think). You can then confirm the dawn phenomenon. You should exclude other reasons first, eg a correction dose or meal close to bedtime, and then as long as you're sure it's your basal night dose you need to bring it down the very next night. You bring basal down by 10-20% at a time - 10% if you're taking less, 20% if you're taking more, therefore you should take yours down by 20%. Repeat your 3am tests until you've got it right. Exactly as you say, a high BG in the morning leads to poorer control throughout the day.
You should do each drop of basal insulin for 48hrs each, so do 2 days at 20% less, test at 3am for 2 days, drop it 20% again for 2 days if necessary, etc. So that you can see a definite pattern. Remember if you're hypoing in the night due to your quick acting insulin still working whilst asleep, then that's something to focus on too. This is because you'd still have a liver dump and it would be due to that rather than the dawn phenomenon. To confirm that, make sure you have your last shot 4hrs or more before bed, then test at 3am, so that you can reasonably exclude it and make sure it's the basal at work.
Good luck. Can you get on a DAFNE course anywhere near you?