LADA is type 1, it's just slow onset type 1 where the honeymoon period may be longer and/or stronger than that experienced in a "classical" type 1 presentation. That honeymoon period might include some time off insulin.
The test to determine if you have type 1 is a GAD test. That is the test to see if you have the antibodies associated with autoimmune type 1 diabetes. However, it might not be reliable so long after diagnosis. Although, I was tested 4 years after diagnosis and was strongly GAD positive and I have seen positive GAD tests over 12 years post diagnosis. AND a negative GAD test is not definitive that it isn't type 1, about 25% of type 1 diabetics are GAD negative. Only a positive GAD test gives you a definite answer: that it's definitely type 1, so there would be no point doing genetic testing for MODY.