I am not sure of the size of the hole in the center disc, but find a stiff wire that fits properly (welding rod? or heavy duty paper clip?) and make a "U" so that one arm of the U goes into the hole (if the center disc also spins, find the point where it must engage with something further "in" and doesn't spin), and one arm (longer?) goes down the side of the cap along the valve stem. Then gently grasp the cap with a pair of pliers so that you trap the longer arm on the outside of the cap against the cap, and turn it so that the "U" that is stuck in the hole allows you to remove the cap.
From the eBay link you posted, the "key" for removing the caps appears to be just a pin that engages with something in the hole of the center disc and has a handle so you can spin the cap off/on.
Failing that approach, you might consider carefully putting some SuperGlue in the hole to lock the outer spinning tube to the inner cap...
As the center disc bit appears to be brass, and lots of valve stems have threaded brass end-tips, you might not have a corrosion problem, just a tight fit.