A number of English-speakers (including those affiliated with the still very functional Hakluyt Society) still use Ernst Georg Ravenstein's Vasco da Gama's First Voyage (London, 1898). Ravenstein was famously meticulous, as I know from his cartography and work with the Royal Geographical Society. He was also a polyglot, formerly of the Intelligence Branch in London, so this is very much in character. This may be available as public domain by this time, as its distribution was wide. [edit: /u/amp1212 makes important caveats below about Ravenstein's source material and the publication itself, as well as sources for the first armada generally; theirs is the superior answer all round. Meticulous, after all, doesn't mean correct or without bias when we're talking about the Victorians.]

There is one issue in the question, however: you imply that da Gama was in Calicut from 1498 to 1502. This is incorrect. He returned in the early months of 1499. His later visit to Calicut with the fourth India Armada was a totally distinct voyage. He did leave his 'ambassadors' behind in 1498/99, and they were quite soon killed, which is part of the reason he sought to punish Calicut and engaged in certain outrages in the process. For the fourth Armada, there is a wider range of accounts in which it appears wholly or partially; if you check Wikipedia for the sources and chronology you can find a start there, in a variety of languages.

The principal source the description of "Christians in Calicut" that I think you are alluding to is not first hand, but it is contemporary. It's the letter of Girolamo Sernigi, a merchant born in Florence, residing in Lisbon. In a letter to an associate in Florence, he writes

The most illustrious Lord Manuel of Portugal sent 3 new vessels to discover new countries, namely 2 vessels (balonieri) of 90 tons each, and one of 50 tons, in addition to which there was a ship (navetta), of one hundred and ten tons, laden with provisions. Between them they took away cxviij men, and they left this city of Lisbon on July 9 1497. Vasco da Gama went as captain of this fleet.

On July x 1499 the vessel of 50 tons came back to this city. The captain, Vasco da Gama, remained at the Cape Verde islands with one of the vessels of 90 tons in order to land there his brother Paulo da Gama, who was very ill. The other vessel of 90 tons was burnt because there were not people enough to navigate and steer her. The store-ship also was burnt, for it was not intended she should return.

So note -- he's writing this without da Gama himself's return, so some of what he's reporting may actually be third hand. He gets to the bit about Christians which is what you see referred to, quite often, but as to reliability-- well its second hand at best

Let us return to the above-named city of Chalichut, which is bigger than Lisbon, and peopled by Christian Indians, as said. In this city are churches with bells, but there are no priests, and the divine offices are not performed nor sacrificial [masses] celebrated, but in each church there is a pillar holding water, in the manner of the fonts holding our holy water, and a second pillar with balm. They bathe once every 3 years in a river which is near the city. The houses in this city are of stone and mortar, in the Moorish style, and the roads laid out and straight as are these

included in A Journal of the First Voyage of Vasco da Gama, Ravenstein ed. Don't be confused by the inclusion in this volume-- this is a contemporary second hand account (if Sernigi was speaking with someone who actually saw this), or perhaps a third or even fourth hand account (if he heard from someone who heard from someone).

Christians were frequently intrigued by the possibility of uncontacted Christian Kingdoms in unexplored lands, notably the "Prester John" story . . . that lead to frequent exaggeration of ambiguous signals; "a pillar holding water" isn't exactly convincing evidence of Christianity, in the absence of priests or religious services. The Portuguese were particularly interested in such stories-- they often locate Prester John's "kingdom" in Abyssinia, there's an interesting looking study "The Portuguese and Prester John of Abyssinia" by Diogo Jose Pereira Andrade [Lisbon: 1940] on the subject. It is just a few years after da Gama that King Manuel is apparently investigating the possibility of alliance with an erstwhile Christian King in Abyssinia.

For a critical discussion of Ravenstein's edition, and some of the controversies surrounding, see

“The 'Roteiro' of Da Gama's Voyage.” The Geographical Journal, vol. 12, no. 1, 1898, pp. 20–21. JSTOR, www.jstor.org/stable/1774495.

An admittedly cursory review of the literature suggests that there hasn't been a lot of digging on this subject in the last century. There would likely be Arab and Indian texts that would add something to the story of Calicut at the time of da Gama ("Calicut" is no connection to "Calcutta/Kolkata" by the way). This was a major spice trading port, and there are other places where accounts might be (including Chinese), were an enterprising scholar with appropriate language skills to look. We have a bit from ibn Battuta, but there must be more in Arabic. There are a number of works on the "Zamorins of Kerala" (the hereditary Hindu monarchs), which I see referred to, and might be further reading.

But while the voyage is fascinating, and there are interesting accounts to read, the sourcing for the common story of Christians that you reference is a weak one, at best second hand.