Not sure if you have some of these details already, but just in case they might fill in a few gaps in the timeline...
There's a 1922 birth showing on FreeBMD of a Kevin Beary in West Derby registration district which is in Lancashire and covers parts of Liverpool. Mother's maiden name is Cribben, and three years earlier in 1919 there's a possible marriage of the parents in the same district - a Eugene Beary and a Blanche M Gribbin (with a 'G') with the same references on the marriage index. Certs can be ordered from the UK GRO (abt £9 STG) which would show more details - shout if you need references and details on ordering.
It seems from Probate listings that Eugene died 8th August 1953 at Northern Hospital Liverpool, without a Will, i.e. an administration, and his widow, Blanche Mary is mentioned. Eugene's usual residence is shown as 60 Edge Lane-Drive Liverpool, and on FreeBMD his age at death is reported as 58, so born c1895.
There's a possible match for Eugene returning to England in 1931 on the Cunard ship Scythia. He's listed as a Salesman with about the right age, and an address at Edinburgh Rd., Kensington, Liverpool.
There are also details of him arriving in New York in April 1929 on the Samaria, from Liverpool. This shows Eugene as born Clonmel I.F.S. (Irish Free State) and age 33 which fits, with occupation Grocer's assistant. Also with him on this journey are wife Blanche and three children ranging in age from 3 to 8, including Kevin in the middle. Blanche's place of birth is shown as Douglas I.O.M. (Isle of Man) and all three children born Liverpool.
I dont see a likely BMD Index reference for Eugene's birth here in Ireland at the moment, but will check further...