Can't find any Rotton in Digbeth in directories. I think it might be a cloths shop, not a butchers, anf figures are models, as they seem to be standing and not hanging. There was a game and fishmonger who wa sRotton, but he was in Dale End. Teh fact that two separate people chose to paint/draw the image would seem to mean it was real. Probably very early 19th century, when directories were very "holey".
Added after thought:
Can't be all that early else there would not be gas lamp