Give us some numbers for the day, including when you wake? Sounds like the Dawn Phenomenon. If you have a bad case of DP as many of us do, then rather than fiddling with overnight basal, it might be better to take a correction dose on waking.
But to plan anything like that, you'll need to test very thoroughly for two or three days and look at all the numbers. Thoroughly = on waking; before meals, meal + 1 hr, + 2 hrs; in any gaps if weird things are happening.
I take 1u or sometimes 0.5 correction dose when I wake up, because my overnight basal is great but my liver greets the dawn by saying, Oops, Lucy needs some energy and dumps glucose on me. It's sweet really. But if I don't, very hard to stop the rising.
That's just a personal example - you'll have to check it all out.
It sounds as if you are going to take better care of your health than your advisors. You will have a better grasp of how your insulin metabolism works. Lots of us work think it's worth working quite hard to keep sugars below a certain level.
As you're a late-onset type 1, why not head over to the LADA forum. You'll find lots of others with similar stories.
Lucy
PS You'll also see many of us weighing up the cholesterol/statins issue. Have a read.