Hi Nobby,
There are huge holes in the apparently "complete" BMD indexes. Someone wrote a book about it (A Comedy of Errors by Michael Whitfield Foster - admittedly focussing on the marriage registeres but I'm sure many of his comments would hold true for both birth and death registers).
With a reverend for a father, I feel sure her birth would not have gone unrecorded. Your best option may be to contact the local registrar's office (this would appear to be the New Forest office, Public Offices, 65 Christchurch Road, Ringwood, BH24 1DH.
Tel: 01425 470150. Fax: 01425 489685) and ask for a search for a certificate, giving her father's name and likely time frame. However, that is more expensive than just ordering the basic certificate.
Otherwise, a baptism record might be an option, and for that you need some kind soul going to the records office in Winchester. Do you know which church her father served?
Mmmm, just looked at his biography in the Clergy List and he was a curate at St Matthias in Birmingham 1857-58 and then was at St Mary Magadelen Taunton 1858 onwards. Neither near Hampshire! How wide did you spread the net? Might she have been born in Leamington for example? Enumerator's error?
Just some thoughts.
Nell