Halberdiers suffer from pathetic morale, very slow speed and are useless in the desert IMHO. Which is why I almost never train them. The best anti cavalry flankers available to the catholics are Chivalric Foot Knights. You can get these simply by dismounting Chivalric Knights on the battlefield. I'm sure that some of the various Crusading Order Knights dismount to CFKs as well, I'm sure Hospitallers are the only exception as they dismount to HFKs.

Arbalests or Pavise Arbalests are still the best missile units to field against armoured units.

The best catholic faction for anti mongol defence are the English. Billmen and Longbows are pretty much designed to take down heavy cavalry and when combined with Arbs and CFKs the enemy doesn't have much of a chance.