https://pubshistory.com/Warwickshire/Birmingham1915/BirminghamBeer1915DE.shtml
Lists two places on Dartmouth Street "Ernest Dalton, 163 Dartmouth street" and "George Henry Doyle, 130 Dartmouth street". I disregarded Grand Junction for the length of the name and because pjmburns says it was on the corner of Chester Street and there's no corner the other side in the picture.
https://pubshistory.com/Warwickshire/Birmingham1892/BirminghamBeer1892M.shtml
has "Mrs Sarah Moseley, 145 dartmouth street".
https://pubshistory.com/Warwickshire/Birmingham1911/AstonDuddeston1911.shtml
has "130 Dartmouth street, Beerhouse, Bowsher108 Dartmouth street, Beerhouse, Void & Uninhabited".
If anyone has access to a trade directory from 1892/1911/1915 they could probably find out where the coffee shop and Dartmouth Stores (Sales or whatever it was called) were. That might lead to a name and address for whatever the pub was in that directory.