Ordinary bows are probably not that good against kats, but arbalests, longbows and javelins etc are pretty good (not sure if they are all AP but they feel like it).

Italian spears should be able to hold the kats for a while, then you need something to kill them. Either a rear charge with your own knights or some infantry with AP. Halberdiers, even urban militia in a pinch.

Sometimes the Byzantines are just too strong to defeat in a single battle. In that case, whittle down their heavy cavalry in successive battles. Missiles are perfect for this - your whole melee army (cheap spears etc) can rout, but if you micromanage it, your missiles can keep firing until the bitter end. Often you can buy back your missiles, but the kats stay dead.