Well this has been mighty impressive - this has been an issue the family have been working away at for 20+ years with various cousins taking a run at it then passing the baton to other cousins to have a go at and it has eventually been passed on to our children. We weren't confident that the mystery wouldn't end up being bequeathed to our grandchildren! Now it looks to have been solved in less than 24 hours!

A*try
Liverpool, England, Catholic Baptisms,
Gulielmus BARON born 31 May 1872
Baptism: 16 Jun 1872
Parish:Our Lady of Reconciliation de La Salette, Lancashire, England
Father:Thomae Baron
Mother:Elizth Barry
That is a great find, the 1939 Register really helps us confirm this - I know a cousin has been through the Liverpool Catholic baptisms before but with a surname of Baron and without the birth date it'd be so easy to miss.
1871 A*try
Thos BARNS occ:fire man born 1845 Ireland with wife Eliza daughter Ellen and sis-in-law Julia BANY(sic).
3813/10/14
That helps nail it all down, it not only confirms the information on the Vicksburg, but also slots in nicely with the great work you did connecting the Barrys to the Lynchs:
Just some thoughts:
James LYNCH marries Mary CONNOR 1904 Liverpool (registrar attended)
Mary CONNOR daughter of Patrick and Julia CONNOR.
Julia BARRY marries Patrick CONNOR Sep Qtr 1871 Liverpool
Julia BARRY daughter of John and Johanna BARRY has a sister Eliza b.~1844.
Possible marriage for Thomas BARNES and Eliza BARRY Jun Qtr 1869 Liverpool.
It is good to discovery that a collection of old memories and rumours turns out to be right (give or take a Berry/Barry) but the odd thing is business of the stepmother. The timing is awfully tight as Thomas would have had to remarry between 1872 and 1875. Also the widow Elizabeth from Ireland with William Barnes in 1881 would seem to be the same person as his mother:
https://familysearch.org/pal:/MM9.1.1/XQD8-X8TThe only odd thing is she is 33, which is quite a bit younger than Eliza Barry (who was 9 in the 1851 census and 17 in the 1861 census and 26 in the 1871 census). I suppose it is always possible that Eliza died and Thomas married a younger Elizabeth but the only relevant death I can find is an 1878 one for a 35 year-old Elizabeth Barnes:
http://www.freebmd.org.uk/cgi/information.pl?cite=UMDluMLnWpHZ83iG%2F9%2BZPw&scan=1There is another wedding of a Thomas Barnes to an Eliza/Elizabeth in 1873:
http://www.freebmd.org.uk/cgi/information.pl?cite=LUTwGMVWCqE03nKITzU5rg&scan=1This appears to be an Elizabeth Hughes:
https://familysearch.org/ark:/61903/1:1:NJJJ-1XPIt is possible that Eliza dead giving birth or soon after and their death was never registered - the family clearly have form for it.
As said here:
Looks like both of William's sisters died - Helena Maria Elizabeth bp.1870 died as Ellen Mary E. Mar.qtr.1872 Liverpool - Elizabetha bp.1875 - if Eliza died Dec.1875, if Elizabeth died Jun.1876.
There is an Eliza Barnes who died in 1875:
https://familysearch.org/ark:/61903/1:1:2J22-7XJIt doesn't matter that much, as we seem to have the key bits of the family pinned down which is more than I was hoping, but it'd be interesting to unearth this element of his story. It might be I'll have to order up one of those death certificates. It could always be that his "step-mother" was perhaps an auntie who came to look after him (it happened on my mother's side of the family). We'll see.