I've come late to the thread
Moving Traffic Contraventions in London in the context of "sign overload" at
Rivercourt Road. I was intrigued by the OP's question "can ‘special authorisation’ signs be used for moving traffic contraventions under the 2003 Act?"
The sign which is at the heart of Merton's signage and which they use to allege a contravention is diagram 618.3C,
TSRGD 2016 Schedule 8 Part 2 Item 2. This was not in TSRGD 2002 but, in accordance with DfT practice, was held in reserve as a sign which they rolled out with special authorisation when someone asked for it. Having tested it as a special authorisation sign for some years, they then made it available for general use in TSRGD 2016. They seem to have used this practice repeatedly over the years: in the late 1980s/early 1990s they tested diagrams 953 and associated signs before introducing them in TSRGD 1994.
In accordance with TSRGD 2016, the signs which Merton use omit the bottom panel (which shows waiting restrictions). TSRGD 2016 provides various options for the Except panel, which include "permit holders" but not "authorised vehicles". I do not know whether Merton obtained special authorisation for this. I consider that, however, to be a minor detail.
The main issue is the one raised by the OP: is the sign one for which moving traffic PCNs can be issued? As a London authority, Merton's moving traffic PCNs are issued under London Local Authorities and Transport for London Act 2003.
Schedule 3 of this Act lists the signs which are scheduled for the purposes of section 4, i.e. contraventions which can be the subject of PCNs. As this schedule was written for TSRGD 2002, it's not surprising that it doesn't list diagram 618.3C.
Unfortunately, legislation.gov.uk only has LLATfL Act 2003 in its enacted form and one needs to check subsequent legislation to work out what the current version is. I searched for primary and secondary legislation since then which included the words "London Local Authorities" and within them for "2003". There are no SIs which are relevant and no Acts since 2013, so it's not surprising that I couldn't find any reference to diagram 618.3C in the Acts which do reference the LLATfL Act 2003; these are the LLA Act 2007 and the LLATfL Act 2008.
It therefore appears to me that the answer to the question which the OP posed is "No". The relevant legislation, LLATfL Act 2003, does not define a contravention of diagram 618.3C as being an offence under section 4. That being so, it appears to me that
all of Merton's PCNs alleging contravention of diagram 618.3C were false allegations.
This argument may help others facing PCNs for contravening any of Merton's signs to diagram 618.3C. How the legal system handles past false allegations by Merton I leave to others.