Hi Fremund,
I think I have found John James' BAPTISM at Christ Church.
23 September 1860, John James MURRAY, son of John and Sarah, Newtown, Whitehaven. Occupation Labourer.
Also to the same parents - 16 November 1862 Joseph, and 13 October 1867, Sarah. Unfortunately the birthdate was rarely recorded.
I looked at marriages in the four main C of E churches but no luck with John and Sarah or John James and Ellen.
In 1871 the MURRAYS were living at 29 New Town (Pipehouse Lane) RG 10 5253/22/37.
John/Head/Mar./46 yrs./Dock Labourer, Bn Whitehaven
Sarah/Wife/Mar./46 yrs./Bn Whitehaven
John/ Son/-/11 yrs./Scholar/Whitehaven
Joseph/Son/-/8 yrs/Scholar/Whitehaven
Sarah/Dau/-/4 yrs./-/Whitehaven
Not much luck with DORAN (they may have been RC).
1871 has Bernard DORAN at 93 Middle Row, New Houses (RG 10/5260/57/36) but I haven't found Ellen. Bernard's age was 35 yrs., a coal miner born Ireland.
Wife Ann, aged 32 yrs., born Whitehaven as are children Thomas (

and Margaret (5).
Another family of DORAN's are at 19 Castle Row, New Houses (RG 10/5260/49/20) James (48) and Kate (43)and they have a daughter Elenor aged 7 yrs. (which is about right) I can give you details if needed but if you are certain the father was Bernard I'm afraid it is not the right family.
Regards
Joan