Faster Than Light Communication Using Gravity?

Alas, no, the communication would not be instantaneous. This is a valiant attempt, and it's been considered by many people before, since we're mostly all taught Newtonian gravity before we're taught about the speed of light, and the former is then never treated again prior to general relativity.

What happens is the shift in the mass of A causes a disturbance in spacetime (a small perturbation to the background metric) and this disturbance travels outward at a speed no greater than $c$. It's exactly the same as if you moved a charge collection and had someone far away with an electromagnetic receiver (well the GR equations are nonlinear and complicated, but the same principle is at work).

You won't be able to see the effect immediately, for example if Sun suddenly vanishes, earth will continue in it's orbit for 8 minutes and 12 seconds before it going straight. Gravity travels at the speed of light...