For some reason Ancestry never allows me to sign in and answer PM’s, so can only answer through the forum.
There are two Edith Anne Thomas shown on Free BMD one in 1874 and one in 1875. Both births are registered in the Pembroke Registration District which covers the parish of Tenby St Mary.
I can only find one baptism probably that of the Edith A. baptised in Tenby in 1874, which would suggest that the 1875 entry is a non-conformist chapel baptism.
There is only one established church in Tenby that of Tenby St Mary, there are a number of non-conformist Chapels.
Birth Mrge and Death certificates can be obtained from General Registar at Stockport and can be ordered over the phone.
Baptism Mrge & Burial parish register entries (not to be confused with Birth & Death certificates, Mrge entries in the Parish Registers are an exact duplicate of the Mrge Cert, unlike the others) can be obtained from the original registers deposited at the Pembs Record Office.