Repair faucet on an old bidet

[fixingperson]

I'm working at a house and the owner asked me if I'd be able to repair one of the taps on an older Amercian Standard bidet. I've done some plumbing and sort of stuck my neck out and said yes but now that I've looked it over I have a few questions. There are two taps hot and cold with a regulator (small lever mounted through the back of the rim turns from rim to spray) to direct waterflow to the rim or the jet. These taps aren't typical and I have no idea what brand they even are. What makes them different is under the handle removable with screw, there is a chrome cup sort of thing (about the size of a shot-glass upside down with a hole only large enough to take the screw for the handle that doesn't thread off or pull off, believe me I gave it my best.

I've never seen anything like it. The plumbing behind the bidet is high-end and removing the whole tap wouldn't be an easy job. Barely roomfor wrenches. The unit has a single chrome pipe that rises up above the taps about a foot with a 2 inch diameter circular cap on it. No idea what it does. All the pipes are chrome, no copper visible. Almost looks like a commercial unit. To replace both taps and maintain the existing elborate plumbing looks like it would be a nightmare because of the array of piping and chrome fittings, nuts included.

All that aside if I could find out how to remove this cap under the tap turn handle (clear plastic) I'm sure I could at least get to the valve and seat which now turns but no water will come through it.

Any ideas what this unit is or what way would be the best to go?