What does your trainer say?
Finding the right bit for a horse is not an exact science - there are considerations to take into play about their conformation, but at the end of the day every horse is an individual (as is the person holding the reins) and there may be a Goldilocks phase as you go through your options. Certain bits might be ok in certain hands, and suddenly unpalatable in others - and sometimes the horse’s age and dentistry plays a role too.
There is a tool out there to measure bits called a bit sizer - this tool gets you the exact measurement you need for your horse. Mouthpiece type and ring type play a role too - you may have to size up on certain bit styles, like loose rings.
Most of the TBs that have come through my program settle in bits in the 5 1/2-5 3/4 realm.
20mm is not uncommon mouthpiece size, but that is rather thick - 16mm is not terribly narrow either. 16mm-18mm seems to be the average mouthpiece size from my small sample size of personal horses.
I have a TB similar to what you describe (which, IME, is very typical of TBs - fleshy large mouths and flat palates) who is in a 5 3/4 eggbutt – but then I have his brother who has very similar conformation in his mouth, who is incredibly particular about his mouth and of all things, has gone the best in a single link Baucher.
Try it, see if it makes a difference – and dont be afraid to play around with different sizes and mouthpieces until you get the “just right”.