I think this type of straw/reed hat is very good, and would be stiff.
[I know that doesn't meet the requirements as specified, although I'm not sure why you've made that stipulation]
Also, is the black hat picutred made of straw? [I think it might be]
And I agree, the Arnolfini hat probably is plaited straw (despite me owning the same design in felt) and not just looking at the picture.
Any hat that expensive in felt would probably be made using a collapsing block/former to allow the taper, otherwise it's too uneven for that posh, and the earliest evidence I know for those is Tudor. [But I could be wrong.]
Anyway, mine is only semi-rigid [not the first time I've said that]
and is more vulnerable to damage in tansport than your average hat.
I have an interest in medieval "top hats", and I do wonder about the construction of hats like the one shown on the grotesque at St Peter's Church at Winchcombe, or Albrecht of Bavaria on the Tomb of Isabella of Bourbon.
[both I think will Google].