The Gemoro implies that the word "ki" cannot mean "that".
What about the posuk at the end of the Sedra Vayeshev "VaYar Sar HoOfim Ki Tov Posor" which would normally be translated as "The baker saw that he had explained well".
Someone said that "VaYar" can mean "he understood", i.e. "he understood, because he had explained well" (according to which "ki" means "because" - one of the four meanings), but I don't think this is plain meaning of the posuk.
Someone else suggested that the Gemoro says that _in addition_ to the meaning "that" there are four meanings. Although this is not the obvious reading of the Gemoro (that "Viyiru ki..." cannot mean "they saw that...") maybe this is possible, i.e. the Gemoro does not say that it _cannot_ mean "that" but that you can _also_ learn that "ki" means "because".
It is also interesting to see Rashi in Chumash Bamidbar 20:29, that "if" (Aleph-Yud) can mean "that" (Asher), hence "ki" could also mean "that". Maybe this Rashi agrees with Resh Lokish (4 meanings of "ki") but disagrees with our Gemoro?
I would be interested in your comments in this matter
Mark - Yes, it is certainly true that "Ki" can mean "that" ("Asher", in Hebrew), as you quoted from Rashi on the Torah. (This thought appears in Rashi here, and in Tosfos here as well, by the way -- except that Tosfos says "Asher" is includede in "d'Har" rather then "Im.")
As for your question from "Ki Tov Pasar," Tosfos (DH Ela) basically asks your question. He brings other Pesukim (an earlier one, in fact), and not the Pasuk you mentioned, but the idea is the same. His answer is a very unclear, "perhaps here this meaning does not fit in as well as in the other verses" (i.e., and that is why we explained it as "d'Ha" instead).
What Tosfos means is probably that in the other Pesukim, the Torah could not have left out the introductory statement with the word "Ki..." (meaning "Asher"), because then it would not be clear why the action that follows took place. That is, in the case you mentioned, we would not know that the Sar ha'Ofim suddenly told Yosef his dream only because he heard a good interpretation (and not because he was waiting on line to tell it to him next in any case). The same applies to the cases in Tosfos. In this case, though, the Torah could simply have said "all the people of Israel mourned for Aharon." It would be obvious that they mourned him because of his death, which was described in the previous verse. Why did it have to repeat "and the nation saw that Aharaon died , and they mourned him...". It must be that the word is "Vayera'u" and not "Vayir'u."
By the way, I did not find that Rashi says in our Sugya that Ki cannot be read as one of the other three types of Ki, in our verse. To the contrary, from Rashi Ta'anis 9a DH Ki Meshamesh it might be inferred that it is possible to read this as a different type of Ki, but nevertheless, we are allowed to be Doresh any of the four Leshonos in any Ki. If so, your question does not begin. Certainly, this Ki could mean "that." Nevertheless, Chazal saw to read it also as "d'Ha."
Be well, Mordecai