Well, I cannot see any logic at all in buying the certificate for the marriage to Kate Champion in preference to the birth certificate.
On the birth certificate you will get the names of both parents and the maiden name of the mother
And if its an illegitimate birth then at least you will get the name of the mother.
The only reason would be if you needed information on Kate.
And as I said, Surrey parish registers held at Surrey History Centre are coming online on Ancestry in 2013. Including marriages to 1937, so obviously that will include the one in Reigate in 1897.
See the info on this on the Surrey County Council website
The birth in 1856 was in Camberwell, not St Olave. And it was the only Summerfield birth in Camberwell district in the civil registration indexes until Harry Summerfield in 1866 (and that's an illegitimate birth)
You need the birth certificate really