Money from the tamchoi (tzedakah) is only given for eating purposes; it is not given for the fulfillment of a mitzvah. If a mitzvah must be performed through the act of eating, one may give from the tamchoi.
Based on this we can explain why the Ba’al Haturim explained that the halacha of giving wine to the poor is alluded to in the gematria of the pasuk mentioning the four leshonos of geulah. According to the above explanation, the drinking aspect of the mitzvah of the four cups is alluded to in the four leshonos of geulah. The other reason – the times the word “kos” is mentioned – represents the aspect of reciting the Haggadah and berachos over the cup without actually drinking it.
Thus, in the pasuk of the four leshonos of geulah, it is most appropriate to allude to the halacha that we can give to the poor, for it is only due to that reason that we can give to the poor from the tamchoi.