As
@Sally71 suggests, waking in the 7s is not a major issue although may be an indication that a low basal could help if your levels rise slowly during the night.
What happens to your levels when you get up and about?
I find I experience Foot on the Floor - my body thinks I am fasting so dumps extra glucose to help me getting going until I eat something. I don't have to eat much or even any carbs. I just need to convince it that I am not starving and it will stop the liver dump. As a result, I try to eat sooner rather than later and definitely need something for breakfast even if it is just a handful of nuts.
My understanding is that the usual advice for someone diagnosed with LADA is to take some insulin to try to take some pressure off their remaining insulin producing cells. Depending what level you go to bed on this could assist with your rise and provide more flexibility so you don't have to plan exercise first thing or eat reduced carb meals which much be restrictive to your lifestyle