Nobody has the right to see a GRC.
In some circumstances they do:
From the legislation:
(4)But it is not an offence under this section to disclose protected information relating to a person if—
(d)the disclosure is in accordance with an order of a court or tribunal,
(e)the disclosure is for the purpose of instituting, or otherwise for the purposes of, proceedings before a court or tribunal,
(f)the disclosure is for the purpose of preventing or investigating crime,
(g)the disclosure is made to the Registrar General for England and Wales, the Registrar General for Scotland or the Registrar General for Northern Ireland,
(h)the disclosure is made for the purposes of the social security system or a pension scheme,
www.legislation.gov.uk/ukpga/2004/7/section/22
You seem to think this indicates that the MoJ have no right to know whether someone has a GRC. It appears to me that (d), (e) and (f) say otherwise.
This section appears to be mainly for the use of people (such as employers) who have obtained this information in an official capacity, and prevents them from making this information public. It is not to make the possession of a GRC a secret that absolutely nobody else can know about.