Mogg is a bit confused here I think. The 1840 Act serves to protect Hansard rather than the supremacy of parliament.
The Met Police was already established 10 or 11 years before the 1840 Act, so while this was foreseen, the fact that there would be cameras, Parliament TV etc was not. Access to Hansard at that time was not so simple a process as it is today.
The requested papers are not papers that have already been laid before the house, therefore the 1840 Act does not apply.
Nonetheless, if I'm incorrect about any of that, it is of little consequence. The Contempt of Court Act 1981 protects ongoing investigations using a strict liability rule which makes it a criminal act for any person to reveal information that may cause serious prejudice to the administration of justice, or lead to bias of a trial jury.
Setting aside any bias I have either way about Mogg or Starmer, Starmer appears to be correct on legal basis for refusing to put the information in the public domain.