Well, I wouldn't like to offer any guarantees
as to whether it would fit, especially since I only ever had a modest (but much-used and cared-for) Systemdek II, but these two images from the Gallery
show, respectively an original Systemdek (close cousin to the III) and a Systemdek IV (early version with power supply built-in). You can see that the alloy sub-platter looks very similar in both cases, suggesting either platter would fit.
As to whether the III platter would give the same performance as the original IV platter, I couldn't say. Obviously, Systemdek's designers believed the redesigned platter was an upgrade, though it was only one of the changes they made to produce the IV.
One can't even be sure it would sound better than the acrylic platter currently fitted to your deck, as also shown in the Systemdek gallery. However, it would certainly offer greater rotating mass, which should benefit speed stability. Firm and generous bass was also associated by some reviewers with the high mass platter when the early decks came out. One must assume that the IV's suspension was designed around such a high mass platter, and would benefit from having one fitted; as you say, the change would bring your deck closer to the original. I'm not surprised you haven't found a IV platter, as the deck is pretty rare.
You would have to use a clamp if you don't already do so, as the III and IV both had slightly concave top surfaces, and the record wouldn't properly contact the platter without clamping. This wasn't the case with the original Systemdek, though, as the top surface was flat with an indentation for the record label.
HTH (a bit!)