Hi Phil,
I don't know anything about you - your weight, lifestyle, etc and I am not Type2, but I do low-carb and manage LADA in that way with Apidra and levemir.
The thing that struck me on reading your post above is that you are keeping your carbs to 'under 100g per day'. The trouble is with Type 2 that it is usually largely based around insulin resistance - and you might well be resistant to injected insulin as well as your own insulin. Honestly, in that situation, I'd cut the carbs right back to absolute minimum; avoid ALL starch carb and get your carbs from green veg and salad only. Your tea last night would have been pork steak with salad or vegetables rather than bread and butter. I know it's hard, but many Type 2s need to cut their carbs right back to 50g or less per day initially. When the base line BG levels improve, the insulin resistance tends to improve too - but this can be over a year or so.
Second thing is that one of the best ways to improve insulin resistance is to exercise regularly - not necessarily for long sessions, but regularly. Even with LADA and very little insulin resistance, I find that my levels are greatly improved and my need for basal insulin reduced when I take regular exercise - and for me, swimming for half an hour has a dramatic impact. Could it be that you are more active on the days that your BG is lower? If you are overweight, this contributes greatly to the insulin resistance, so cutting carbs drastically and exercising regularly might help with that too.
As i said above, I don't know anything about you as i don't normally read and respond to Type 2 questions, so please feel free to ignore what i've said if you are already doing these things. I hope you find a way forward soon.
Smidge