Hi former LADA here (officially dx in 2011)
It's a pain but are you able to set your alarm 2am and check your blood sugar?
To see if your basal level is holding steady until you wake up or changing. Reason for 2am is to allow time for your dinner bolus to be out of your system and therefore allowing clearer result of how your lantus is.
I was put on Lantus injected 7am. I perservered 6mths and a ton of jelly babies, the Lantus caused hypos every few hours and was unpredictable sometimes. For some it's not a smooth profile like it is claimed to be. It's definitely 20-24hrs long and especially for LADA types in the early days the 1 daily injection will do.
Levimir however is smoother so you might want to ask for that if you find the Lantus is causing some unpredictability.
You will have to move to 2 doses morning and night eventually but this is expected for the more common basal insulins available. On plus side splitting gives more flexibility especially if you exercise.