@LADA enigma - I would push for a CGM as well - it's the only way you'll get any insight in to what's actually going on in your own body.
On the Thyroid - the hormone itself is (not surprisingly) made in the thyroid, but the call for production comes from the Brain, via the Pituitary - that's what the T4 (hypothalamus) and TSH (Thyroid Stimulating Hormone) is looking for - is the signalling around the thyroid working properly.
Please do not take this as a diagnosis - but look at what is most likely here (and I'm being guilty of comparing to my own story) - C-Peptide is telling you that your pancreas is working - so you are producing insulin - yet you are responding to additional insulin.
My experience is that the medical system is wary about changing diagnosis, because they focus on glucose only. By prescribing insulin, you will control your blood glucose, and everything will appear to be fine (whether you are really LADA or really Type 2). However, if you are producing your own insulin, by definition, you are an insulin-resistant Type 2, and while the additional insulin is overcoming that resistance and lowering your blood glucose, it's doing nothing good for you in the long term.
This puts you in a tricky position, because the safe thing for the medical profession is to continue as is, whereas - you really need to know whether your underlying issue is a lack of insulin or too much. The only way you will understand that is with a CGM. If that sounds scary or confusing - please reach out to your endocrinologist (this is a hormone issue after all) and go through it with him.