Neat idea! Seems like the pulleys need to be fixed into position for the swinging to stop.

Picture taking a bit of ply and making a couple of right triangles. One leg against the wall, another extending out to the pulley position, the hypotenuse following the general line of your current mounting rope. A bit of 2X stock between the two will maintain their relationship to one another. The existing screw holes can be used to mount the thing to the wall.

As long as the angle is such that the pulley is held above a line between the hooks and the point of force, the rig shouldn't require additional hardware to hang stably.

Or, you could hard-mount the pulleys to the wall rather than on the extensions. Then create trucks for the weights to run up and down the wall (think of stealing your kid's roller skates and some duct tape and you'll have the idea.) They then won't have any movement of the rope to cause a swing, and the wall itself will attenuate any swinging that might occur before it has a chance to amplify to damaging proportions. You do risk wheel marks on the wall, but a piece of carpet or such could be hung from the same hooks as the pulleys to effect the result.

Ideally, I'd like to see the pulleys mounted lower for a more realistic force vector. But then you'd need to route up to another set to get the range of motion needed from the weights. Too many holes there. You could mount them all on a flat frame of 2x4's, and simply use the wall hooks as a way to keep that vertical while in use. That also handles the weights banging on the wall, but makes things less portable.

Those are my thoughts. Not sure that any of them are worth even what you paid.