Sounds vey much as though you are sleeping through a night-time hypo!
The only way to find out is to test durning the night, you can either keep testing every hour over-ne night or several nights so that you cab build a picture what is going on with your levels..
Between 2am-3am is around the point that your body is at it's lowest ebb for hormone production and is also the lowest point you bg should fall over night, as the hormones production increase after this point, this stablelises or slightly increase your blood sugars...
There could be several reasons for these hypo's, one being that your background insulin isn't correct, so you have too much insulin over-night.. But I did notice that you mention a low carb tea, I wonder if you are using too much quick acting insulin here..
A good book to get hold of, that goes through carb counting, insulin adjustments also providing information of how to test your background/basal insulin is correct, is Think like a pancreas or Using insulin..