Is hunting permissible with a BB gun (air rifle) using non-sharp, round pellets, knowing that it may kill the game by shock and from a close range, and is this considered religiously permissible hunting?

The hunting tool must be sharp and wound the flesh by cutting or piercing. The air rifle hunts by piercing and penetrating the body with its pellets.

It is stated in "Al-Mawsu'ah al-Fiqhiyyah" (The Fiqh Encyclopedia) that the conditions for the tool are: 1. The tool must be sharp, wounding and affecting the flesh by cutting or piercing. It is not a condition that it be made of iron. 2. It must strike the game with its sharp edge, wounding it, and it must be certain that death occurred by the wound. Otherwise, its consumption is not permissible, because what is killed by the side of the tool or its weight is considered "mawqūdhah" (killed by a blunt instrument).

As for hunting with lead bullets, scholars have differed concerning it. Some of them permitted it due to the piercing and penetration that occurs with it, unlike clay pellets which kill by crushing and breaking. There is no difference between a lead pellet and an air rifle, especially when hunting small animals.