Another probably useless idea: Place the far end of the space grid sufficiently far from the region that you are going to perturb that the signal will not reach it in the time interval you are going to compute over (so if you are going to run the model from 0 to t place the far space boundary out at kct where k is of the order of a few and c the wave propagation speed). Then use a zero boundary condition at the end.
I have first done it this way. But, for my experiments (3D), a grid should be at least 1000x1000x1000xtime in this case, and it is very memory consuming. Your first idea about a transformation is interesting, but I'm not sure how will the different grid size influence the stability.