Short answer would be that your pancreas is still producing a decent amount of insulin itself, which is naturally released overnight (at a rate of about 1 unit an hour if everything is working properly) and that is enough to deal with your (overnight) fasting sugars.

It's time to have a discussion with your prescribing Doctor, and if they are not a Diabetes Specialist, perhaps time for a consultation.

It is possible that your Doc is trying to use a Basil Insulin (Long acting - Lantus in your case) to try and deal with "mealtime" sugars, which may be your greater issue, that would more properly treated with a Short acting Bolus insulin.

If it is not your fasting sugar (liver released sugars while sleeping or not eating) that are your real issue but treating for them, that certainly would explain your morning low AND your daytime highs.

There would also be an oral medication alternative (Sulfonylurea class drugs) that could be taken before meals if your insulin production is decent but just need a bit of help.

Those are my thoughts but I would really "Talk to your Doc" and insist on an answer and explanation. And if they can't give you one, then ask for a consult to an Endocrinologist who is best equipped to sort things out if you don't fit into the "standard treatment guidelines" which all Doc's use.

Lantus commonly results in low blood sugar readings if taken in the morning.Try taking your dose at bedtime.My dose was changed from am to pm whilst in hospital recently for that very reason.Once my sugar was stable again, i changed dose back to am as that was what i preferred(been my routine for more than 10 years)