You intuition is good in that taking an extra 4 units of Abasaglar on a night will likely cause a greater risk of hypos through the night and particularly when you are doing exercise on an evening as that in itself can drop your night time levels whilst you sleep. Exercise can have a BG lowering effect over the following 48 hours.
The rise in BG toward morning may well be what is called Dawn Phenomenon and is a result of the liver pumping out extra glucose in anticipation of you starting the day ahead and needing more energy (a throwback to prehistoric days when we didn't have kitchens with cupboards and fridges of food and we needed to get up and hunt or forage for breakfast), but as
@helli says, your Abasaglar may well have run out by then so your body can't deal with that glucose.
With a waking reading of 7 I don't think it is a concern at the moment as that is a good waking reading, but dipping down to 4.1 in the night is not ideal (just a bit too close to the red line for comfort). What are your daytime readings like? Just wondering if reducing your morning Abasaglar a bit might stop things dropping too low in the early hours but still keep your levels in range during the day.
Great that the Lift tablets and biscuits with peanut butter strategy kept you out of the red last night.
It is common for non diabetic people to find their BG levels dip through the darkest hours of the night and then rise again in the morning, but everyone is different and it can change over time. I was dipping like you are in Nov last year. Now I start going high in the late evening and struggle to keep my levels from going too high. Things rarely stay the same with diabetes and you learn different strategies to cope with those changes.
Basal insulins have a particular profile of activity. Some are more uniform and longer lasting than others, but the body is rarely uniform in it's basal needs, so finding a basal insulin which has a profile which, as closely as possible, approximates your body's need is important. People on insulin pumps can adjust their basal insulin hour by hour to match the amount their body needs at different times of the day. Those of us on Multiple Daily Injections (MDI) have to try to find the best balance with just 1 or 2 injections a day and once it is in, you can't take it out again so if it is too much at certain times of day/night you will hypo. Tweaking the time you inject and sometimes splitting the dose between 2 injections and varying those doses independently can all help or changing to a basal insulin which might have a slightly different profile of release which may suit your body better. These are all quite clumsy adjustments compared to the finesse with which your clever little pancreas used to perform the job, so it is always going to be a best fit rather than ideal.