HaLeiVi: I believe LF was referring to the Abarbanel, not Al Naharos Bavel. And the reason for the Abarbanel was that he considered morale Pikuach Nefesh in that case. Nothing to do with performing a Kiddush Hashem.
To answer the original question (outside of Yome Kippur, Shivasar B’Tammuz, and Tishah B’av), it’s just as Assur or Muttar on any fast day as it is all year round.