We know that Ergo is both a power source for Puppets and a manifestation of their memories and soul, so if the Scrapped Watchman was powered by the Golden Ergo, that would've been absorbed by either P or Manus, as the former absorbs the vast majority of the the boss' Ergo, particularly the portion which contains memories and personality, and the latter absorbed residual Ergo with his machine.
That level of dilapidation shown by the swamp Puppet would be inconsistent with the scale of time in the game. The swamp Puppet had been left in the swamp for so long that any paint (if any) it was coated with peeled away, and there is a clearly a thick layer of rust across its entire body. The events of Lies of P take place over days, and we know the Scrapped Watchman
wasn't moved from the City Hall as late as the defeat of Archbishop Andreus.