A devil's advocate counterclaim would be that no, the Hunting Bow would not benefit.
The benefit comes if it is used to attack "an enemy creature that was attacked and damaged by a friendly animal creature"
The Crab's text says merely: "Colossal Crab's prey receives 2 direct damage" in the upkeep phase. Not, you note, that "the Colossal Crab does 2 direct damage to its prey".
So technically speaking it is not specifically stated that the crab is the cause of the damage, merely that the damage occurs. It could be Wrath of God stuff or something.
(I believe that the Bow would benefit based on RAI, but from a strictly RAW view I don't think a case can be made.)