I am "three sixteenth Battye" as my cousin puts it and work in Huddersfield, although all my Battyes were Holmfirth based.
I have tried to follow up on your queries but need a bit more information. I found the Liverpool entry for Edgar in the 1881 census, but this is not very informative, as you know. I did not find any definate local candidates for Thomas and Mary. However, I did find a possible candidate for Thomas in the 1851 census.
Can you advise me how you know the Mary's surname and whether you have a copy of the marriage certificate? Even a date? I need some way to fix Thomas' age and fathers name, then I may be able to help you.