The court will look for the following facts:
how much alcohol has been consumed;
body weight
time after last drink
the results of analyzer.
Medical condition is not excuse for that
In theory hypothyroidism should take you longer to metabolize alcohol; also for the heavy person more drinks will needed to achieve the same level than for the skinny one.
In my experience, while there are factors that will make you drunk quicker, having Hashimoto's is not one of them, nor will it make you stay drunk any longer than not having it.
I should add you can try anyway with a doctor's letter as this could depend on the judge. The lawyer told my sister told the judge she was seeing never takes notice of letters from doctors.
My sister was in the exact same situation and found that unless the doctor is going to be in the courtroom, a doctor's letter holds no sway with the judge. We live in Queensland Australia just to add.