Firstly, your claim about Stillwell's "conclusion" is a misrepresentation. Stillwell makes no such conclusion about the lack of transmission. In fact, he explicitly states that the Kerala school knew these mathematical series before 1540. This selective reading and distortion of Stillwell's work is intellectually dishonest and undermines genuine historical inquiry.
The Jesuits sent to India weren't not your typical bible thumpers; they were highly trained mathematicians and astronomers with a specific mission to study and acquire Indian mathematical and astronomical knowledge. The primary motivation for Europeans to import knowledge from India wasn't mere academic curiosity - it was a matter of practical necessity, particularly in navigation. By the mid-16th century, Europeans were grappling with significant errors in their calendar calculations. The true solar year was about 11.25 minutes shorter than the assumed 365.25 days, an error that had compounded over centuries, leading to serious discrepancies in timekeeping and navigation. Matteo Ricci, the Jesuit astronomer and mathematician in a letter from India to Giovanni Battista Maffei (Italian mathematician) he states that he requires the assistance of an “intelligent Brahmin or an honest Moor” to help him understand the local ways of recording and measuring time.
If one wants a smoking gun—a direct admission of knowledge transfer—is either naïve or deliberately obtuse. Do you also believe that tea plants magically teleported from China to India? The British East India Company's industrial espionage in China's tea industry parallels the Jesuits' activities in India perfectly. Both were covert operations aimed at acquiring valuable "know-how" for economic and strategic gain. Do we have a signed confession from Robert Fortune or his kin admitting to tea espionage?
The cumulating circumstantial evidence isn't just substantial—it's overwhelming. We have documented records of Jesuits studying Indian texts, teaching Indian concepts, and corresponding with European mathematicians (see my other comment for examples). The methodological similarities between Kerala mathematics and later European work, like the striking parallels between Wallis and the Yuktibhasa (15th century), where Wallis (in 17th Century) is using the exact expression and reasoning as given in the Yuktibhasa, aren't coincidences—they're smoking guns.
Your dismissal of this substantial body of evidence goes beyond healthy skepticism. It appears to disregard the complex realities of historical knowledge transfer risks coming off as a deliberate attempt to erase non-European contributions to mathematical history.