Hi goanna, as far as I am aware if you are LADA you will need to keep taking insulin as LADA is a form of T1 and you will be insulin dependant, no ifs and no buts, if you stop insulin your bg levels will rise and Byetta will not help.
If you are T2 then Byetta is probably a good option for you as you have put on weight whilst using insulin.
As far as I know the only way to tell 100% if you are T1/LADA is either a Glutamic Acid Decarboxylase (GAD) antibody test which tests for Type 1 diabetes antibodies, not present in T2 diabets or a C-peptide test which measures the amount of insulin C-peptide present in the blood. C-peptide is a naturally-occurring by-product of pancreatic insulin and so will be high in T2's and very low in T1/LADA. (As far as I am aware)
Were you diagnosed as LADA after either a C-Peptide test or an antibody test (GAD)?
If not I would ask for these tests to be carried out to definitely define what type you truly are. And if these tests werent carried out then I imagine your doctor can only have been assuming you were LADA. And as the old saying goes, when you assume you make an ass out of u and me, never assume always ascertain.
Good luck and all the best