(This is my first post here in ages and I apologize in advance if I totally garble this -- I'm coming at this from an Early Modern European studies background and I haven't cited anything in a million years.)

The reputation of the Society of Jesus as a threatening, highly politically active organization was firmly entrenched by the time Shogun is set, and certainly by the time it was written. You can hear echoes of this in a lot of modern-day writing about the Counter-Reformation -- the image of the Jesuits as "the Pope's stormtroopers", the shock troops of the Catholic church, etc. To hostile sources, the Jesuits were identified by their willingness to sign off on assassinations of rulers and regime changes, their duplicitousness (including the use of rhetorical sophistry and their willingness to order to lie under oath), and their rigorous blind obedience to their own hierarchy. In Europe, Jesuits were frequently seen as diabolical representatives of a foreign power -- shadowy cat's-paws of Spain and Portugal, both Catholic countries semi-unified under Philip II, and indelibly linked with the interests of those countries over the interests of their individual nationalities. In Japan, Jesuits were representatives of European influence, period, but their conversion efforts extending to middle- and upper-class Japanese individuals were seen by many as an effort to get a foot further in the door for Spanish and Portuguese interests in Japan. Portuguese traders were very willing to include such goods as firearms and European-style armor alongside decorative goods and confectionery; the first hand-held matchlock arquebuses were introduced to Japan in 1543. This wasn’t exactly gun-running in the modern sense, since these weapons weren't broadly considered contraband, but it would have definite political consequences in a politically volatile and sword-heavy nation, and the link between Jesuits' religious interests and their financial interests (especially via silk trade) is not negligible. You even get later Jesuits complaining that other flush-with-cash Jesuits in Asia have lost sight of their commitment to poverty -- in 1573 Francisco Cabral, superior of the Jesuit mission in Japan, celebrated the massive financial loss of the sinking of a Jesuit trade ship loaded with silk as the occasion for a return to proper Jesuit attire. (Check out Linda Zampol D'Ortia's "Purple Silk and Black Cotton: Francisco Cabral and the Negotiation of Jesuit Attire in Japan (1570–73)".)

Francis Xavier, one of the earliest members of the Society of Jesus, first visited Japan in 1549. The Society of Jesus was highly active in mission work throughout the 16th century, including campaigns in India and China as well as the New World; Jesuits readily adopted the Japanese language and Japanese customs for their purpose of propagating Catholic Christian thought, including writing Japanese-language catechisms and devotional tools. Basically, if you want a European organization to completely take the plunge in entering a non-European culture and trying to engage with it on at least an approximation of its own turf, the Jesuits are your guys. Moreover, their strategy involved taking advantage of secular governmental structures and forging strong relationships with political and social influencers, rather than distancing themselves from political life. In Europe, this meant accepting positions in close proximity to already-Catholic kings and princes as well as an active role in arts and education; in Japan this meant strategically proselytizing to the warrior aristocracy and taking advantage of preexisting feudal loyalties to disseminate Christian allegiances. This isn't to say that Jesuits in Japan only pitched Christianity to elites or that this was a cold-blooded political maneuver; this was a strategy to secure the benevolence of local rulers and to facilitate more broad-level missionary activity among vassals. In areas where this top-down strategy didn't take, Jesuit missions often failed to take root as well; it's hard to effectively proselytize in an area you're plainly unwelcome in.

The initial Japanese willingness to entertain Christian proselytization was linked to trade interests and to contemporary power struggles within Buddhism, but after the first flush of expansion and mass baptisms relations became strained. In addition, as information about the way Catholic mission work often went hand in hand with European colonial expansion filtered into Japan, local rulers understandably balked at the prospect. (This wasn't a mere coincidence or an unfounded suspicion -- while individual Jesuits might not have been consciously dissembling their colonialist intentions, you can look at the New World or the Philippines for what this could look like, but it's way out of my area of expertise so I won't even try to touch it.) Interactions between Japanese individuals and Jesuits were variable to begin with, and the latter half of the 16th century was marked by shifting opinions of Christianity among local daimyo and the violent suppression of Catholic agents in Japan, not only Jesuits but competing Franciscan missionaries as well. While initially tolerated, these representatives of Catholicism and their converts came to be perceived by some ruling daimyo as a threat to the unification of Japan and as hostile to traditional Buddhist and Shinto practices, including the destruction of shrines.

This ebb and flow of political opinion culminated in events like the public crucifixion of 26 Christians in Nagasaki in 1597 after prominent daimyo Toyotomi Hideyoshi (who the novel fictionalizes as Nakamura) took the shipwreck of a Spanish vessel off the coast of Japan as an overture to an invasion of Japan. In 1600, when the novel Shogun is set, Japan was in an state of political instability verging on civil war due to conflict between regents after the death of Hideyoshi -- contemporary Jesuit documentation identifies Tokugawa Ieyasu (the novel's Yoshi Toranaga) as the preferred candidate and likely victor. They expressed hopes that the death of Hideyoshi marked an end to persecution of Christians in Japan and would open up opportunities for policies friendlier to Christian expansion and renewed mission work in the territories possessed by preexisting Christian lords.

So Jesuits clearly had a dog in that fight, and both Jesuits and their allies and non-Christian daimyo sensed that 1600 was a turning-point year for the presence of Christianity in Japan. There were at least fourteen baptized daimyo by that time, with varying levels of Christian observance. But the Jesuit optimism proved misplaced; in the ensuing conflict of that year several of these Christian daimyo were killed, apostatized, or otherwise ceased to be in a position to tangibly support Catholic missions. When Tokugawa Ieyasu took power, rather than being sympathetic to Christian interests, he took note of the lingering threat of invasion by Spain and Portugal and would ultimately implement an edict for the expulsion of Christians in 1614. Violent suppression of Christians only became more decisive in the 17th century, culminating in Japan's near-total break with European trade altogether and the complete expulsion or coerced deconversion of any remaining representatives of the Catholic Church.

Later in the 17th century, with Spanish and Portuguese traders effectively locked out, Dutch Protestant representatives were permitted to remain in a diminished capacity, restricted to a limited geographic area and forbidden from proselytizing. This is super interesting in its own right but probably not relevant to your interests.

Amazing. Thanks!