What gives? Was red dye (from Brazilwood) that much more in fashion in France than in Portugal at the time?
Well the answer is going to be long. Firstly, politics, you had a dispute between the Atlantic navy and the Mediterranean navy. Second, the Atlantic navy was very decentralized and this would provide the funds needed to reform and truly centralize that navy. When in otl this was only possible centuries in the future. Third french pride, France never accepted the Treaty of Tordesillas ( with it´s king asking to see Adam's will that would exclude France from the division of America), Brazil has been for France the land of all commerce: ""commerce of souls" for Protestants or Catholics ( from Fort Coligny, in Rio de Janeiro, to Fort Saint Louis, in Maranhão), "product trade" by smugglers, pirates or privateers (in Paraíba, São Vicente or Rio de Janeiro for Dugay Trouin, for example) and " trade of ideas” by scientists and artists (from La Condamine to the French Artistic Mission of 1816).
Every trip to Brazil was extremely profitable. Brazilian wood, in addition to being used for clothing, musical instruments and furniture of extreme high-quality , were used to dye paper sheets in the Rouen industry. Not just brazilwood, but animal skins, fruits and exotic seeds. Live animals such as monkeys and parrots (animals highly valued in Europe). Concern for France was so great that the Portuguese king, at the beginning of the 16th century, published a decree ordering all his subjects, under penalty of death, to sink all French ships leaving or arriving from Brazil. And that was not without reason, in 1540 nine ships from Rouen were send to Brazil; fifteen Dieppe ships and ships from Brittany are equipped for the same destination (24 in total). In 1541, thirty to forty ships left for South America and especially Brazil. In 1546, a fleet of twenty-eight ships left Le Havre bound for Brazil.
Fourth reason was geographical position, Brazil's vast coastline presented a very strategic advantage for France. It provided access to the Atlantic Ocean, which was crucial for trade and naval power projection. Not only that, but it would allow French privateers to attack any enemy ship that passed the Slave Coast. Disturbing Iberian Asian trade and preventing (or at least greatly increasing the danger) of expeditions from rival nations. In addition, in the future they could hinder the sale of slaves from Africa to the Caribbean or demand a protection tax. Also the control over Brazilian territories would give France would allow the establishment of further colonies ( In africa and asia for example).
*Parmentier complained that the King of France (did not take seriously the influence of the French among the inhabitants of Brazil: "If the King of France had wanted to free himself a little from the French traders in less than four or five years, they would have won the friendship and assured the obedience of the people of these new lands, and this without any other weapons than persuasion and procedures”. For the great captain, in the space of these four or five years “the French would have penetrated further into the interior of the country than the Portuguese in fifty years , and the inhabitants would probably persecute the latter” * - Barbosa, Mario de Lima, 1923,
Les Français dans l’histoire du Brésil, Rio de Janeiro-Paris, Briguiet & Blanchard.
The idea of colonizing this region comes primarily from Nicolas Durand de Villegaignon, born in Provins in 1510, gentleman and soldier, knight of Malta and vice-admiral of Brittany, whose classmate was Calvin at the University of Orleans. In 1553, he pleaded with Gaspard de Chatillon-Coligny, admiral of France, for a project to colonize Brazil. The latter quickly understood the political interest of such a colonial expansion that would consist in dealing a blow to Iberian hegemony in America, which easily convinced Henry II: “A young and strong France would emerge overseas" . Basically the idea of colonizing Brazil became a popular trend among the naval nobility and traders. It was the thing of the moment, the topic that never went away, a fever about a grand French empire over the world that would begin with the colonization of Brazil.
A curious thing was that while the Catholic French colonists mixed with the natives and had mixed children, the protestants (Calvinists and Lutherans) vehemently refused to have relations and contacts with the natives. A French philosopher and historian in the 20th century said: "French Calvinists who were all inflated and triumphant, tried to create an exclusively white colony in Brazil and left the country without leaving traces of colonization."
The French spelt Brazil with a "z", not and "s"?
Brésil
(edited after the message) PS: The intense French maritime activity is accompanied by numerous acts of piracy and aggression against the Portuguese to try to expel them from the region, both in Brazilian waters (the first French ship was boarded in Brazil in 1504, in the bay of the river Paraguaçu), as well as at mandatory crossing points on return routes (Azores, Ireland Sea and other important connections). The situation got so bad that the French Atlantic merchant and military navy was attacking any Portuguese ship they encountered on the way to Brazil. Basically this navy (France had two for each ocean, the Atlantic and Mediterranean) was in an open war with the Portuguese navy (even if France was not) That is why, in a spirit of appeasement, the French and Portuguese sovereigns, after signing a treaty of friendship in Lyon (July 14, 1536), they convened in Bayonne, in 1537, a Court of Awards, intended to resolve conflicts between French and Portuguese shipowners. This Court, however, did not bring any agreement, except that Francisco I undertook to prohibit his subjects from trading in Brazil. The Atlantic navy pretended to obey, but continued to act aggressively towards the Portuguese. This is something repetitive in this era, the Atlantic navy did not always obey the king. An gran admiral of the Atlantic navy even said that after the king he was the most powerful man in France. This is due to the search for financial sustainability of this navy, which was secondary to the Mediterranean navy and the fact that the ships did not belong to the king but rather to nobles who "lent" these ships to the king. So theoretically, if this group becomes stronge enough, the king will have to deal with a parallel state using the colonial empire to strengthen itself against the Mediterranean navy and royal decisions against their interests.