A slight possibility for the name might be the Peter Walker & Co brewery. This should be distinguished from Peter Walker & Son, which later became Tetley-Walker and part of Allied Breweries. Peter Walker & Co bought the Willow Brewery at Wrexham off Robert Evans and had some pubs in Birmingham. The only one in Birmingham I have record of is The Rainbow in Bordesley High St (https://www.breweryhistory.com/Defunct/WM_Birmingham.htm), but the Midland pubs site says (https://www.midlandspubs.co.uk/breweries/staffordshire/peter-walker.htm) that they had pubs (presumably meaning more than 1) in Birmingham. Possibly those referred to previously were two of them. The brewery was taken over by Atkinsons, and I don't know about the Small Heath Tavern, but the George can be seen to be Atkinsons in the photo. A bit dubious an explanation perhaps, but was the Smallheath Tavern Atkinsons also ?