Somehow you don't reflect, what the quote says:
Dummett in a footnote:
Footnote 39 reads: "Gregory 1940, p. 157, quotes this letter, asking the Duke to send to Sigismondo Malatesta a pack of "carte da trionfi de quelle fatte a Cremona", together with a straw hat, another speciality for which Cremona was famous. She quotes it again in Gregory 1958, p. 32, where the phrase is given slightly differently as "quelle carte da trionfi che se ne fanno a Cremona".
Whatever information Gregory had, according this Cremona is famous for straw hats, not Fermo (I don't mind, that Fermo ALSO was famous for straw hats).
The relevant deck with straw hat ...
... was made after 1450 (at least that's the general assumption cause the use of Sforza coins) and a little more precisely 1451 OR 1452, a suggestion based on an letter from Bianca Maria to Francesco Sforza ... we would like to see this letter, but this seems difficult ... in which it seems somehow clear, that Bianca Maria reflects a request of Sigismondo Malatesta for Trionfi cards from Cremona. The date is insecure, as some speak of 1451 and others from 1452 and the most precise, what we've heard, is November 1452.
So there's the suggestion, that this letter refer to one of the decks with the straw hat at the magician table (aka Pierpont-Morgan-Bergamo-deck or better 14 Bembo trumps).
Somehow Gregory has captured from the letter (unknown to us), that a straw hat was involved in the activity (second item send to Malatesta). The reason is unknown, maybe ... if it's correct ... that it was simply harmless and meaningless accident.
For Fermo and its straw hats we have, that Francesco Sforza was thrown out of the city by a rebellion in winter 1445/1446. Sforza went to Pesaro from there.
For Cremona we have, that it is assumed, that Bembo produced there the relevant Tarot cards ... with straw hat. Cremona also was famous for straw hats (according Gregory). Sforza had been variously in Cremona after 1446. The deck is assumed to be produced 1451/52, not "before 1446".
The engraving with German hat (nobody says, that's it is a straw-hat) was made c. 1480, possibly reflecting iconographic Italian influences on German-Flemish artists ... that's not clear. It only repeats the hat feature, perhaps only, that these these Bateleur artists generally occasionally used a hat for some table magic.