In a collision of two billiard balls, is there a threshold energy that is required in order to create a black hole?

No, classically there is no threshold energy for such a process. In an asymptotically flat spacetime, the ADM mass(-energy) is conserved, and classically there is no minimum mass for a black hole. Therefore if the pool balls each have mass $M$, and in addition there is kinetic energy $epsilon$ in the center of mass frame, then the collision can lead to the creation of a black hole with mass $2M+epsilon$. The value of $epsilon$ can be small or even zero without violating conservation of mass, so there is no kinematic threshold for this process.

Of course, there are dynamical reasons why this will not happen. For example, if we know the mass and Young's modulus of a pool ball, then we can estimate the compression of the balls in the collision. This does not result in compression to anything like the Schwarzschild radius of a black hole with mass $2M$. But this is a dynamical argument contingent on the details of the process, not a kinematic threshold.