I thought the same about kaffarah since that is what I had read in Bahishti Zevar but brother ylulat sourced Nur ul Idah and Radd al-Muhtar, where a detailed explanation is given.
My understanding (must have read this somewhere) is that there is Ikhtilaf inside our madhhab regarding the issue of kaffara for different Ramadan fastings; so the most correct thing to do would be to ask a scholar which one is the mufta bihi opinion..
And on a side note, there are various non-mufta bihi opinions in Nur al-Idah..