Struggling with the 1900 US for them and Illinois doesn't seem to have an online birth reg index covering that time

Re your census info Replies #14 and 16
The 1881 suggests that Daniel had remarried (1881-1891) and (slightly) changed his occupation:
1881
Ash St Cottage, Ash next Sandwich
RG11/993/12/17
Daniel Beer, 35, Baker and tailor
Eliza, wife, 36, b. Canterbury
Alfred J, son, 11
Minnie, d, 8
Annie M, 6
Austin, 4
Victor C, 1
All but Eliza b. Ash
So Alfred's mother was likely to be Eliza and not Mary.
gnu