Today's daf quotes Abaye, without any contradictory opinion, that "Kara Li Mo'ed" in Eicha refers to the extra day in a malei Tamuz in the midbar--i.e., what we would call the first day of Rosh Chodesh, but in reality erev Rosh Chodesh. In hilchos Tisha B'Av there are a few minor changes in the davening because the same word "Mo'ed" in Eicha is applied to Tisha B'Av itself. Are the two drashos ompatible?
Good question. The most simple answer is that the word "Mo'ed" in the verse cited by Abaye (Eichah 1:15) refers to Rosh Chodesh, while the word "Mo'ed" in another verse (such as Eichah 2:7 or 2:22) refers to Tisha b'Av, as is implied by the context. In the Halachos of Tisha b'Av (TUR OC 559 and SHULCHAN OC 559:4), the source for the practice not to say Tachanun on Tisha b'Av is the MORDECHAI (Ta'anis #635, 636) and the HAGAHOS ASHIRI (Ta'anis), and although they say that the reason is because Tisha b'Av is called a "Mo'ed," they do not quote an exact verse.
Otherwise, it may be suggested that the two Derashos are indeed from the same verse; the Mordechai's Derasha is based on the simple meaning of the verse (although it is not the way Rashi learns it) while our Gemara's Derasha is based on the "Drash."
Best wishes, Mordecai