I venture to add a few additional reasons why I doubt that Karski spoke of his visit to a camp in the first days of December 1942.
Let us first recall that when Karski was interrogated by the British at the end of November 1942, the report did not say a word about a visit to a camp.
Adam Puławski, in his very interesting article "Revisiting Jan Karski's Final Mission" (Israel Journal of Foreign Affairs, vol. 15, 2021, issue 2, p. 289-297,
https://www.academia.edu/57800603/Revis ... al_Mission ) says:
"Karski himself had suggested in January 1943 that thePolish leadership abandon the emphasis on persecution and terror in occupiedPoland and instead emphasize the ferocious resistance being mounted against the Germans so as to galvanize Allied support for Poland's political goals. (...) It was no coincidence, therefore, that on December 30, 1942, in an interview with Frank Savery, an employee of the British Embassy to the Polishgovernment, Karski asked whether “the Allies were aware that Poland was not only suffering, but also struggling,” and it was in the context of joint struggle that he mentioned Jews: “The entire Polish population of Warsaw, in which ahandful of surviving Jews may be included, is united not only in hatred of theGermans, but also in resistance.”
In February 1943, Karski met Anthony Eden, British Foreign Secretary. Eden asked Karski about the fate of the Jews, but (see again Puławski) Karski was more interested in talking about the conflicts between Polish communist and non-communist resistance fighters. Puławski's text suggests that Eden found nothing of note about the Jews except this statement by Karski: "The Polish population of Warsaw, including the remaining Jews, is united not only in hatred toward the Germans, but also in resistance."
If Karski had agreed to clandestinely visit a camp with the specific intention of testifying to the West about German atrocities, is it not surprising that as soon as he arrived in England or thereabouts (December 1942), he avoided to talk too much about the sufferings of the Jews and that, even when questioned by Eden about these sufferings, he did not recount his visit to the camp?
Wood and Jankowski 2014, p. 284, 3d note on p. 186, say: "Karski's tales of atrocity. In his last report before leaving England, Karski noted that the stories wre part of his standard presentation in his second round of meetings with Britons". This report is not dated, but Wood and Jankowski say it is from May 1943. So there were two sets of interviews with Britons and it seems that it was only or mainly in the second set that Karski told his stories of atrocities.
It would be interesting to know the exact terms of this Karski report. And by the way, there is another report, dated March 25, 1943 and signed J. Kwaśniewski (another pseudonym of Karski, after Wood and Jankowski) on conversations with English and American personalities in London (Wood and Jankowski 2014 , p. 275). This report would also be interesting to know in detail.
For the above reasons, it seems doubtful to me that in the first days of December 1942 Karski told Schwarzbart that he had visited the Belzec camp. One can wonder if at a certain moment, he did not re-endorse the testimony of another courier. (We know that several Polish couriers traveled simultaneously between Warsaw and London, see Puławski's article.)
Of course, it is possible that Karski behaved with his Jewish interlocutors differently than with the Allied politicians.
In any case, I think it would be good to have a clear idea of Karski's reports kept at the Hoover Institution Archives.