Hi I'm T1 and not T2 but I have lots of experience with hypos through the years. Cgms (continuous glucose monitors) made a big difference to me when they came out, because they can be set to warn you
before you go low and they give you a reading on your phone. They are expensive though and I have my doubts that you'll get one on prescription if you're only on one injection a day.
As a diabetic you'd be entitled to a free trial of the libre
Claim your 15 day free sensor if you have diabetes and have not tried the FreeStyle Libre 2 plus sensor before
and the dexcom
Dexcom One Plus Sample Programme
I've got to wonder if you'd be better off on a slightly different insulin regime so that you don't have to take carbs to feed the insulin rather than taking the right insulin for your carbs, but suspect that that might necessitate multiple injections a day. I take it your doctor didn't want you to reduce the insulin to reduce the overnight drop?
Can I ask what insulin you are on?