Electronic – RSSI to distance moving average

There is a configuration of two coils, known as a Helmholtz coil, that is supposed to produce a nearly uniform field. The idea is to have two coils that are separated by a distance about the same as the diameter of the individual coils. Going with this, the coil diameter would obviously have to be about the same as the length of your rails, so would be rather large relative to the rest of the apparatus.

But you might not need an exactly uniform field anyway. If you had a field that was non-uniform, but symmetric across a plane that divides the mid-line of your rails, the forces on the movable bar should be in-line. It might be possible to produce a field like that by using two rectangular coils, same width as the rail separation, with one half above the plane of the rails and the other below.

A ferromagnetic ring will sit and vibrate at the AC cycle rate. Pulsed DC is likely what you need. Typically "washer launchers" are a big bank of capacitors with voltages in the kilo (kill-you) range that are triggered to dump and "fling the ring". If you do not have the equipment to handle these measurement tasks, that is where you need to start. Get the test equipment.

For non-ferromagnetic, but conductive, rings, you sound like you're on the right track.

There is a physics youtube channel that has a "launcher" of the sorts that is sounds like you are talking about. They use room temperature [Al] and test, then the same ring Liquid[N] cooled.

Keep in mind the device in this video uses UK line voltage.

The Electric Cannon