Looking at the 1881 census, Elizabeth seems to be Henry and Elizabeth's eldest child - and if a birth record is hard to find for an eldest child it's always worth considering that he/she may have been born before the marriage, and so will appear in the birth index with the mother's surname.
In the 1871 (taken on 2nd April) census unmarried Henry Barber aged 18, porter born in London is living in Bermonsey with his mother Martha. (Henry in 1881 is 28 and a warehouseman)
(Class: RG10/ Piece: 628/ Folio: 47/ Page: 28)
Is this your Henry? If it is, then Elizabeth aged 10 in 1881 must have been born before he married.
There is a possible marriage of a Henry Charles Barber with Elizabeth Margaretta Talbot in June quarter of 1871, in St Saviour (Southwark) district.
A birth of Elizabeth Margaretta Drane Talbot was registered in Thetford district in 1851. Elizabeth Barber in 1881 gives her birthplace as Brandon - which is in Thetford district.
I'd be happier if I could find Elizabeth Margaretta in between her birth and her marriage, but have had no luck so far. You might not want to do this, but a birth certificate for one of the later children of Henry and Elizabeth Barber would tell you for sure what her maiden name is.