I am actually a labor and delivery nurse who is diabetic and has been for 28 years.

I am now 15+5 weeks pregnant and I also have been having terrible lows with my problems mostly during the night as well. Unfortunately for us, it's going to continue until probably in the mid 2nd trimester, around 24 weeks. That is what my doc are able to tell me anyway, but with as with everything it all comes down to each person is different and it can be different for each person.
I can tell you have a pump because you were talking about your basal rates-do you have the continuous glucose sensors as well? I just got them about a week ago and so far I like them, although they can be annoying with how often they alarm. That may be something for you to look into.
Do you see an endocrinologist? If so, maybe what you need to do is on a weekly basis, fax in your blood sugars and have your doc go over your results. He/she can then make adjustments for you so that it isn't so bad. That's what I have been doing, although I still seem to have problems getting totally in control of everything. I am feeling a bit frustrated because I think I am gaining more weight because I have to constantly be eating snacks to make sure my sugar is staying normal.

I just wish that I could be having a normal carefree pregnancy!!!!! Good luck to you!