I think you've nailed the location as being the High Street Vivienne. But maybe not quite on the corner with New Street. - At that point was a coaching Inn called The Swan - one-time haunt of Samuel Johnston. - It originally had a courtyard entrance for the horses, but was remodelled with the hotel set back from the road.
Here is an etching of it in 1829 - which is within a year of when the sketch may have been done...
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You can see lanterns of exactly the same type as shown in the sketch and a bollard near the corner. In the sketch you can see in the extreme bottom right-hand corner that the kerb is turning inwards - This could well be the coach entrance into the Swan hotel. The hotel was later reduced in size and the coachyard developed into a narrow road known as Swans Passage.
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In this etching from much later (1887) - You can see the Sign for "The Swan Hotel" on the roof of the buildings on the right - But that was just an advertisement for the hotel which was behind them in Swan's passage - The passage came out near the bollard next to the lamppost where there is a kink in the pavement. The bollard may even have been one of those shown in the sketch, slightly repositioned
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You can imagine the sketch being drawn by someone staying in the Swan Hotel in 1830 - Maybe having arrived by stagecoach. Popping out early in the morning to do a quick sketch down the high street, and then wandering down to do another sketch in Bell Street (the sketch in the link given by Morturn)