I have a copy of an Oxford journal from the 19th century that contains a collection of records compiled by John Stow when he was commissioned by King James in 1604 to start a systematic historical survey of London. Stow had already established his reputation as London's first historian of note and this royal patronage gave him nominal access to any subject's personal library and the right to interview whoever he chose (and get paid for it). Alas for the impoverished Stow he was too old and infirm by this time, and died only a few months later, but not before he had actually set out to do the king's bidding. This brief period of research never yielded any publication, but in the 19th century was made available to scholars when purchased by the Bodleian (I think, it might have been a college library).
In the journal article Stow lists some infamous murders, one of which happened outside The Pie on Thames Street in the time of Henry VII in which a clerk was robbed and stabbed. Stow conjectures that the clerk, being in the employ of Morton the Archbishop of Canterbury, may well have been on an errand for his master at the time and that fate therefore could easily have led any fellow young clerk to his doom that day. If so then the great Sir Thomas More could well have been lost to history before he'd even reached maturity.
It is unlikely that there would have been two Pie taverns in the vicinity of the docks - the point was to keep the names as unique as possible within general areas.
Thames Street is a barren concrete wind tunnel these days, devoid of character, devoid of aesthetic quality, devoid of daylight for much of it. It is certainly devoid of taverns - though a few hardy licences have managed to withstand the ravages of post-war development, just about. The Pie, alas, had long passed into oblivion. The docks, including Queenshithe, were meat and drink to the Great Fire of 1666 because of their highly combustible warehouse contents and many a tavern in the area went up in smoke along with everything else. If The Pie had managed to survive to that year it definitely does not reappear after it as far as I can see, though I see that a tavern still existed across Thames Street from Queenshithe dock right up to World War Two - so maybe this was its successor (London always had a remarkable manner of retaining tavern locations over impossibly long periods in which buildings, names and clientele changed many times over - an even better retention rate than church locations in some areas).