Well, I don't pretend to have any big answers here
and this is ONLY my opinion. However, I AM hypothryoid and take medication for it (half of my thyroid has been removed because of a tumor) so am somewhat knowledgable in that aspect of things.
First, sounds like you need a new doctor if yours is an unsupportive as you say! Regardless, in my opinion this is NOT something you should do without having good medical supervision.
I googled First Cleanse and looked at several sites. First off, most of the easily available info is very vague... and they have prominent statements on the sites indicating that their claims have not been backed up by the FDA. One might wonder why not!! In fact quite a few sites claim it is good for "stimulation of thyroid metabolism." There are even claims you should do this instead of needed surgery or medication. But again, NOT backed up or approved by the FDA! Do the words "snake oil" ring any bells?
And if you are on thyroid meds, doesn't seem wise to take something else which claims to affect the thyroid function.
Again, this is only my opinion. But I certainly would not do it, especially with the thyroid meds. But then I wouldn't do it otherwise either. But we DO need to take our health problems into consideration and when we have a medical condition we need to be exceptionally cautious lest we take something which could interfere with the proper working of our medication. If done at all, it should be under a doctor's close supervision.
At least that is what I think.