Another Joseph Smith publican https://pubwiki.co.uk/Warwickshire/BirminghamR/RedHouse.shtml
I think this one is too old.
GRO references:
1886 Q4 Death - Joseph Smith; age 72 (born c1814); Birmingham; 06D; 95
1892 Q2 Death - Mary Smith; age 78 (born c1814); Birmingham; 06D; 81
Censuses:
1861 Rose Inn, 49, Great Russell Street, Birmingham
Joseph and Mary
Ref RG09; Piece 2163; Folio 20; Page 33; Schedule 188
1871 348, New John Street, West, Birmingham
Joseph and Mary
Ref RG10; Piece 3126; Folio 63; Page 13; Schedule 75
1881 348, New John Street, West, Birmingham
Joseph and Mary
Ref RG11; Piece 2999; Folio 22; Page 37; Schedule 194
1891 348, New John Street, West, Birmingham
Mary, widowed
Ref RG12; Piece 2391; Folio 119; Page 33; Schedule 193
(Sorry, can't provide links to FamilySearch records as FS appears to have taken it's ball home in a sulk).
Probate index:
Joseph Smith
late of the "Red House" Inn New John Street West Birmingham
licensed victualler and maltster
died 25 Dec 1886
at the "Red House"
Admin 21 Feb 1887
granted to Mary Smith of the "Red House", widow the relict
Estate: £33,307
https://probatesearch.service.gov.uk/Calendar?surname=smith&yearOfDeath=1887&page=54#calendarProbate index:
Mary Smith
of the "Red House" inn New John Street West Birmingham
widow
died 08 May 1892
Probate
to James Alfred Foster, chemist
and Edgar Williams, officer of excise
Effects £17,469
https://probatesearch.service.gov.uk/Calendar?surname=smith&yearOfDeath=1892&page=24#calendarBirmingham Daily Post
29 Dec 1886
page 8
Deaths
Smith
- On the 25th inst.,
- in his 73rd year,
- Joseph Smith,
- of the Red House, 348, New John Street West
- deeply lamented
Birmingham Daily Post
14 May 1892
page 6
Deaths
Smith
- on the 8th inst.,
- at The Red House, 348, New John Street West, of this city,
- aged 79,
- Mary Smith,
- end her was peace
There are number of newspaper items that link Joseph to the Rose Inn/Tavern throughout the 1850s but I couldn't find a census reference for him.