Billmen are indeed among the best foot units of the game due to their availability cheap price relatively high armor piercing attack and bonuses versus cavalry all with a solid morale, decent armor, decent defence and decent stamina.
The Byz. should have no trouble in getting them as long as they have Varangians that are in a different league altogether - however in late you have to do without them. Your best heavy melee units are indeed cavalry (byz cav/ pronoiai - kataphraktoi are too vulnerable to the many good ap units of late due to their speed), and unfortunately Bills can handle them because of the +3/+1 bonus against horsies and the ap attack.
Your foot melee units (urban militia and byz infantry) cannot match the bills unfortunately, so all you are left with is tactical tricks: shoot them with bows and crossbows; tire them by using your inferior melee units to grind them and then cavalry charge them to brake them; pin them using upgraded spears in hold position/formation and charge them in the rear/flank. Use strong chargers - pronoiai if possible.
Also muster whatever decent foot mercenaries you can get your hands on - they are imperative for Byzs in late; have Inns in your border provinces (that mercenaries are attracted to) and kep an eye on billmen, chivalric men at arms and other such overpowered infantry units as well as for almohad urban militias and even militia seargents.
If you want to peranently solve the problem of Byz foot units in late, conquer Switzerland and pump out Swiiss halbs that are a really good poison against all other poisons.

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