I do not believe we can ever rely on our Livers helping out. Just one sniff of Insulin in the body and it just wont happen anyway. I am fortunate though to not have to worry about it as I do not believe I have ever had a night time Hypo.
Reducing the amount of carbs should also mean a drop in Insulin which may mean the Liver helps out more to begin with but the body should adapt to using stored fat instead.
I have been playing with low carbs for 3 months with 50g to less than 80g a day. I feel better for it now but the first couple of weeks I felt very tired and lethargic. I have now reduced my Basal Insulin by half and my Bolus by much more. The trouble with me is I am not strict enough with it so probably have not ventured too far into the Ketososis phase although I did lose over a stone in weight.
My BG levels used to fluctuate wildly with varying highs between 8 & 15, probably due to my bad management but with low carb the average was around 6 with whole days of between 5-6 readings including night time. I have never been able to do that before.
This you tube video is a discussion with Dr Ian Lake a T1 from age of 30 who is an advocate of low carb diet. There are so many interesting points in it. The main one for me is that only 10% of T1's in the UK reach a Hba1c of 7% or lower whereas those on a Keto diet it is 90%!
He also talks about his (and others) recent "Zero 5 100" event. No food for 5 days running/walking 100 miles. Around the 40 min mark he says that many of those taking part were regularly showing BG of 3 with no adverse effect or any of the usual Hypo symptoms.