Hello Stuart,
I reckon your problem could be the set up of your auto enriching device.
I don't have a PHD in Kugelfischer set up but I have stripped and rebuilt the auto enriching device on my Flavia 1800 injection. I didn't have the special tools that the manual mentions when it came to installing this back on the engine and setting it up, so there was a fair bit of trial and error. During this, one thing I discovered was that moving the enriching device lever up and down by hand, when the engine was cold, altered the idle speed. You can't really move the lever by hand after setting the device up as when the engine is cold, the lever is held in the raised position by the adjusting nut on the push rod - if that makes sense.
I know you said it's working fine but maybe it is not set up quite right and this could account for the wildly different idle speeds. You've obviously noted when your engine is hot the push rod of the auto enriching device extends. Have you noticed if it extends enough to allow the lever of the enriching device to lower all the way down, effectively turning it off. At this point, the gap between the adjusting nut and the lever should be around 3mm at 80 degrees C. If the gap is noticeably more or less than this when the engine is hot, then the adjusting nut/lever will not be returning to the correct position when the engine is cold and this could account for the idle speed being wrong when cold.
God knows if this is making any sense...
So, I would get the engine up to full operating temp, with the idle at 900 rpm ish, and then check the gap between the adjusting nut and the lever. If it's not around 3 mm then you could undo the lock nut and alter the adjusting nut until the gap is around 3mm. Hopefully when the engine then cools, the enriching lever will be in a new position that produces a more acceptable cold idle speed.
This is how I set mine up and the cold idle speed is only a little faster than when the engine is hot.
I will try and post a picture to help with this.
Cheers, Rob