The suspension components (springs, shocks, and bump stops) are different for the three versions of the 500. The springs are colour coded to identify them. Playing around with them as you suggest might work, but you are taking on the role of development engineer at this point and it could prove a costly and futile exercise. This is because each power unit and gearbox combination has a different weight, and the springs and shocks are tuned to this weight. So, if you fitted the setup from the 1.3, you might find the suspension is actually harder because the 1.4 engine does not weigh as much as the diesel.
What you really need to improve things are softer, dual-rate springs, firmer shocks to control them, and probably a rear anti-roll bar to keep things in check. Pretty much what Ford have already done, I suspect. But Ford do not offer the 1.4 engine in the new Ka, so there is no suitable set of parts available off the shelf.
As far as ride quality goes, the shock absorbers will only be part of the problem. They are probably cheap and nasty, and do not control rebound very well. But the initial fidgety, harsh response will be caused by springs which are too stiff in the first 50mm of movement, and the nature of low-profile tyres (as you have found for yourself by unwittingly reducing tyre pressures).
It could be that one of the sophisticated coil-over kits (like the KW one) would actually improve ride quality if you kept the ride height the same, rather than lowering it, especially if it had dual-rate springs. However, it worries me that they are selling exactly the same kit for all variants of the 500, so one wonders just how well developed it really is. And they are not cheap.
John