If I made a hole in the center of the moon I would experience zero gravity because all of the mass would surround me. Since a black hole is so named do we consider that the center also has zero gravity?
At the center of mass point, you are saying that you would experience zero gravity. What is correct to say is that you would feel weightless.
When r=0 the Christoffel symbol Γrtt is zero and that means the radial four-acceleration is zero and that means you're weightless.
Now your question for a Schwarzshield BH does not have an answer, because at r=0 the gravitational acceleration is undefined.
Anyhow, the point I'm gradually getting around to is that your question simply doesn't have an answer. This is because if you are asking about the idealised Schwarzschild black holes then the geometry is undefined at r=0 so the gravitational acceleration is undefined there. If you are asking about real black holes then the gravitational acceleration at the centre is finite because they haven't finished collapsing yet, and indeed won't finish collapsing at least until an infinite time has passed, or indeed not at all if they evaporate due to Hawking radiation first.