In reply to you question, what is the difference btn CAT3SINGLE and CAT3DUAL. THE a320 has two auto flight channels, each with different components affecting it. These components can be the Flight Director(s), ATHR (auto trust), AP (autopilot(s)), RA (radioaltimeter), FMGC etc, as well as flight controls, hydraulic systems, electrical systems, DMC (DIgital Management Computer(s)), FMGC etc.
These components play a role when it comes to each channel's operation. When one system-component becomes inoperative the entire channel drives off line for a full integrity auto land. This is when the integrity of the entire system is affected. This is what we call capability degradation because we can still autoland but with only one channel without the other complementing the operation and backing up. When one channel is off line because its components are affected, then we are operating an approach in CAT 3 SINGLE status. When no component is affecting any channel, then we are operating under CAT 3 DUAL because both channels are online.
When we have both channels on line (CAT 3 DUAL) then the autopilot as a whole can afford the failure of a component that affects a channel without compromising the integrity of the system below 100' RA (Alert Height). When we have only one channel online (CAT 3 SINGLE), then the autopilot as a whole system cannot afford the failure of a component that affect a channel without compromising the integrity of the system at any point during the approach and downgrades the airplane capability to CAT 2 or even CAT 1 (depending on the importance of the component affected or the number of components affected). Back to CAT 3 DUAL, when a component that affects a AP channel fails, the system becomes CAT 3 SINGLE unless, UNLESS we have flown down past the 100' RA mark (Alert Height), below which the AP system can afford losing any channel-affecting component without downgrading the airplane capability. That is, in CAT 3 DUAL, you will NOT see or hear (triple click) a capability degradation should ANY failure occur, including an engine failure or an autopilot failure.
In short, what CAT 3 DUAL does for you that CAT 3 SINGLE does not do, is it guarantees you an auto land if you have a failure in the most critical region of a LVO ILS approach (below 100' RA)