Is there a limit to how fast spacetime can bend/warp around an accelerating object? If in the future a spaceship can accelerate to nearly the speed of light, will it encounter any kind of resistance/drag from the spacetime in front of it not being able to bend/warp fast enough around it?

It is my understanding that as an object accelerates, its mass will increase and this increase in mass should result in the object having more gravity, and this gravity will continue to increase as the object's speed increases. The creation of more and more gravity should result in an increased area of spacetime surrounding the object being bent/warped.

So, if there is a limit to how fast spacetime can bend/warp around an accelerating object, along with an increased area of spacetime being affected by an accelerating object's increased gravity, will this spaceship start to heat up and begin to melt the faster it accelerates?
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.