I think that would depend on the speed at which the bowling ball is moving across the trampoline. Because the stretching of the fabric takes time. If the speed of the ball is great enough, a complete field around the ball (symmetrical depression) wouldn't have sufficient time to fully establish itself at every position along the path of the ball. You would have non-concentric "stretching waves" bunching up in front of the ball and loosening up behind. Sort of like sound waves from the horn of a fast moving car (i.e. Doppler Effect).
Then again, since the stretching waves are to be analogous of gravitational waves, the rate at which they propagate would be light speed and constant. And so perhaps a symmetrical depression (gravitational well) would fully establish itself along each point of the ball’s path?
Hmm, I don't know : ) This whole relativistic stuff is not always immediately intuitive. It probably depends on which frame of reference, keeping in mind that the speed of light is a constant, as you well know. Analogies aren't always perfect.
For some odd reason I had always thought that clocks on orbiting satellites ran slower than ours on earth (because they were traveling faster). But it turns out that our gravity well and the acceleration of clocks in that gravity well caused our clocks to run slower.