I don't see any other Jane Nary or Neary birth registered 1877-8 which might have been the child baptised 13th Jan. 1878. The Jane Neary aged 16 on 1894 ship's manifest would have been born around 1878, if she gave her true age.
A Patt Nary was registered in Castleconnor District, Dromore West Sup. Reg. District 1866. Born 22nd Nov., Castletown. Parents Paddy Nary & Bridget (Loftus), abode Castletown. Paddy's occupation was herd. This would fit the age of the Patrick on the 1894 ship manifest. He wouldn't have been the only Patrick Nary/Neary of a similar age though. So the Patrick who went to U.S. in 1894 had been there once already, returned to Ireland, then left again after a stay of less than a year. Is there any clue as to his destination?
There were 22 Nary births registered in Dromore West 1864-1890 and 30 with surname Neary. Those registered as Nary with parents Pat & Bridget included Andrew born May 1865 and an unnamed male born Feb 1869 who may have died shortly after. None of the Nary/Neary girls registered in Dromore West had parents Bridget & Patrick. Bea's mother may have been born before civil registration, or in another district or the parents didn't get around to registering her. Other possibilities are that she was a child of a previous or later marriage of Pat or Bridget or that they went to Britain for a while and she was born there. Birth registration indexes for England & Wales can be searched with a combination of baby's surname plus mother's maiden surname on GRO Index or on Free BMD. Free BMD is compiled by volunteers and is incomplete.
*No civil registration marriage of a Bridget Loftus to a Patrick Nary/Neary in Ireland. If they married in Ireland it was probably before 1864. There may have been 2 couples Pat Nary/Neary and Bridget Loftus in same area around same time as Shane pointed out. If you draw a blank with parish registers, check for a marriage in Britain as another possibility. People went back and forth for work, sometimes as casual workers but some stayed in one side of the Irish Sea for a few years. *See PS.
One death registration to consider is Bridget, wife of Patrick, aged 54, Castleconnor district, Dromore West, 1897. Ages recorded at death in Ireland at the time were not always accurate and in the cases of old people in particular could be out by a decade. E.g. there was a Patrick Neary death registered in Castleconnor District 1893. His age was recorded as 93 but that could have been a guess. Bridget and/or Pat may have moved to live with relatives and died in another district.
Other female names beginning with B: Belinda, Bertha, Beesey/Beeza. I think the last 2 are Bridget.
If Bea's eldest child was 15 by late 1935 then Bea would have married 1920 or before. You can search marriages on Irish Genealogy site for that period. Try without Bea's forename in case she had another name. Search British marriage records too. As well as the sources I've mentioned, Scottish records are on Scotland's People website. Bea's husband would have had to go wherever there was work, if he was fit. Unemployment was high in Britain in 1930s and there was the Means Test for men who applied for the dole (unemployment allowance). Bea's children's births were too late for you to search on Irish Genealogy. GRO Index for English & Welsh births also has a cut-off date of 100 years. However, some volunteer sites, local indexes and those on pay sites have indexed some births in England & Wales after 1918.
*Edit. PS. I've re-read Shane's post again, properly this time and seen the 2 Patrick Neary/Nary & Bridget Loftus marriages 1861 and 1877. So which children belonged to which couple? Patrick, the son who was 28 on the voyage to U.S. in 1894 was too old to be from 1877 marriage. The Jane born 1878 could have been from either marriage. Patrick's occupation was herd on some birth registrations. Were either Patrick or Bridget on a census?