LADA does come on slowly. It's a slow-onset form of Type 1 basically.
Remember that the blood glucose meters have a margin of error, so a 3.7 could be, say, a 4.5. In addition, my consultant told me that people without diabetes sometimes have blood sugar in the high 3s.
I don't have Type 2 so can't comment on that, but some Type 2s are of normal weight and still have Type 2.
Good luck with your HbA1C test. Testing for antibodies would be useful too, but bear in mind not all people with LADA have antibodies. You could also ask for a C Peptide test which can give an idea of how much insulin you're producing.
Even if your HbA1C is ok, if ask for it to be repeated in a few months just to be sure.
Do let us know how you get on