TyroPicard, thanks for your input, but I disagree. The SD is very representational. The Single Engine + APU config, as you rightly point out, power each side, but the bus ties are still actively able and ready to transfer in case of GEN failure. This is the context of QRH 5.04 and is not considered split, even though on the SD page it looks like they're split.
There is a complex reconfiguration of electrical supply during a CAT III DUAL approach. JETZ Tech explains the nub of it. The autopilots and other essential systems for the approach must be completely independent and isolated to provide redundency. The bus ties powering these systems must be kept open. If there is a failure of one generator the autopilot and other essential systems for the approach on the failed side would be dead. The reason the failed half is not completely electrically dead is because of the unique bus configuration in CAT III Dual.
I know it could be argued why do we need to know this, but I fail to see why we as pilots shouldn't be told this. We know if it says CAT 3 SINGLE on the FMA not to do a CAT 3 DUAL, and we don't start pulling CBs willy nilly because we're told not to. My theory is that a shallow FCOM is less to do with some devilishly ingenious French philosophie but more to do with the hassle of translation. Like No Joke and many others here I'm not so good at sticking my head in the sand.