Think it was the building circled on the market pic.
Don't know when it was demolished

It was demolished in the mid-1920s when work began on the construction of the Mersey Tunnel, because the building had a deep cellar that was close to the proposed route of the tunnel and it was felt it presented a risk to the tunnel's roof. The building nearby on the corner of Albion Street, opposite the Glass Barrell, went for the same reason, as did the block on Market Street between Hamilton Street and Douglas Street, though the latter only went in 1928, when they showed signs of subsidence. That's the reason none of these sites has never been built on since, because the tunnel is so close to the surface at this point.

The building circled is actually the one demolished to build the tunnel, on the 1950s map above it would have been the curve just above the sentence 'Kings Square Entrance'

Quite right. I was paying too much attention to Derek's photo of the site of the Market Hotel as it is today, instead of to his old pic of the Market Square. Having looked at the full version of the original postcard, the Market Hotel is unfortunately not visible on it, being hidden by the corner of Market Street and Hamilton Street. Still, my account of the end of the pub still holds, when a photo of it does eventually turn up.