Hello,
Yes I do have a similar situation to you. My bloods will be at their highest before bed and remain that way (or sometimes creep up) into the early AM. I inject my long acting insulin (Levemir) at 10pm and see my bloods are steady or rise slightly until about 1am-2am where they gradually decline until I wake up. Personally, with Levemir - I've noticed it starts working after about 3-4 hours. This would explain the occasional rise in BGL until about 1am-2am.
A BGL drop from 11mmol/L to 3.9mmol/L is quite substantial though. If you mention this to your diabetes team they may suggest that you try and aim for a figure of around 7-8mmol/L before bed and maybe use less background insulin to avoid such a large drop. But I wouldn't make any changes before discussing this with the professionals.
I hope this has helped some.
Grant