www.nationalarchives.gov.uk/catalogue/ has a couple of naturalisation files for Kluths but nothing that would appear to be Christian.
I see in freebmd his name is indexed as "Clouth" (and she is listed as Martha Jane Baker, not Barker), at the marriage. If you have this certificate, the witness names might help locate him on the 1881 if he was there. You could also look to see if there's a naturalisation record under the name of his father.
Naturalisation status on censuses is notoriously inaccurate. Just "British Subject" etc without "Naturalised" could mean he was born overseas but his father was a British Subject, or that he was naturalised, or that the census was wrong. Many people didn't bother going through the process.