When crafting dispensers, the game is insisting on the use of undamaged bows.

This includes both the actual crafting (when ingredients are placed on the crafting grid) and the game's calculations as to how many dispensers can be crafted from available ingredients. It does not seem to matter if I'm placing items into the crafting grid manually, or if I'm using the recipe from the book.

If a damaged bow is used in the crafting grid, the output slot is blank.

Additionally, I've noticed that when I have one undamaged bow in my inventory and I pick up a damaged bow with the mouse, the tally of how many dispensers can be crafted increases from one to two; but as soon as I put the bow back into an inventory slot the tally goes down to one again.