The bonus damage you get when fighting in berserk without much armor on ("you lust for blood!") does not apply when you are not wielding a weapon. This might be fine to leave for most classes, but Beastfighters, at least, should get the bonus when fighting unarmed.
Monks also can't True Berserk - that is specifically coded in, and makes sense. Beastfighters (and everyone else) can True Berserk, but only when wielding a weapon - and wielding a weapon gives them big penalties. It also doesn't make sense that some random class that opts to fight unarmed can't True Berserk either.
No, you see, this is intended. A normal sentient creature is not able to channel the pure, murderous energy required for true berserking without making use of a manufactured tool of murder. Beastfighters can't do it either since that murderous energy is incompatible with the natural innocence of the beast. Without that weapon, you can't tip yourself over the edge.