Is there a limit to how fast spacetime can bend/warp around an accelerating object?
Changes in the bending of spacetime propagate through spacetime at the speed of light. If I wave my hand thus the (very, very, very tiny) changes to spacetime will propagate outwards as gravitational waves at the speed of light and won't reach the Sun for 8 minutes.

No matter how massive an object or how great the acceleration, spacetime elsewhere only "feels" that acceleration after it has passed into the acceleration's lightcone.