Brian,
Here are some dimensions to hopefully help you out.
There is no minimum differential as any movement will cause some action. There is a maximum of about 5/8 inch differential however. If you go beyond that the supply side of the Slinky will lift off the platform and that is under static conditions. With a 3/8 differential, there is about a 30:70 ratio of Slinky on the platforms. When we play with them, our hands hold the last few coils on each end and this prevents "lift off". Unless it is operated very slow, I suspect that some mean will be required to hold each end to the platform. A little glue or some mechanical means like a thin metal clip screwed to the platform maybe. There is a lot inertia in the Slinky coils. Maybe tilting the platforms slightly toward each other with a shallow pocket for the end coil would work.
For reference, my Slinky measures 2.688 OD and 2.490 ID. There is a little crimped on metal clip that secures the last turn and it is 0.160 wide by 0.160 long. Made out of 1/32 sheet stock. With out securing the ends, the best spacing seems to be about 3/8 to 1/2 inch between the OD's of the coils, IE about 3-1/16 to 3-1/8 inch center to center distance on the coils. I really think you will need to secure the ends of the coils to the platforms and in that case the distance between could be increased and I think that would look a little better.
Hope this helps.
Gail in NM,USA