My first piece of text is related to Companies House registry, you are right.
HMRC doesn't have publicly accessible companies register...
...However I couldn't find them in the publicly accessible register neither here (the service cease on 31 of March 2013) nor here.
go here
https://customs.hmrc.gov.uk/msbregister/search.do?type=newSearchthen in "Money Laundering Regulations registration number" field enter 12713897 (I took it from their site, you quoted it as well)
in Postcode type EC1N 8SB
click search
then you will see a record but for another spelled company "Coin floor" (with a space, not Coinfloor) and with another address.
my screenshots from the previous post show it clearly