Which blood tests should I get to rule out LADA?

A few months ago I had an A1c of 6.0. I don't fit the profile of type 2: extremely thin, exercise daily, ate very little sugar, no family history of diabetes. I understand it's still very possible for me to be developing type 2 but I want to rule out LADA.

I went on a keto diet and my A1c dropped to 5.5 after about 4 months but my fasting blood sugar is still usually in the 110-120 range. My doctor now will not order tests to rule out LADA because my A1c has dropped out of prediabetic range.

So, I am considering paying out of pocket and doing it myself at Walk-In Lab. But I am not sure which tests to order specifically. Does anyone know? Thank you.