There really is no need to use a bench for any of this. The "hanging" part of the back raise can be handled by dead lifts, good mornings, those type of exercise where you raise to an upright position against resistance.
The only part of the exercise you describe that isn't worked can be worked satisfactorily simply by laying on your stomach, and raising your chest off of the ground, keeping your legs and hips flat on the floor.
Another possibility, although space may be an issue, is to use a Swiss ball for the same motion. Work on balance and stability first, as the ball is more mobile than a bench.