As one might expect, there's a number of old pubs called "The Dragon."

F'r instance:


The Dragon, Ilkley, West Yorkshire (est. 1642)
   
The Dragon, Lymm, Cheshire (est. 1670)
   
The Dragon, Market Harborough, Leicestershire (est. 1714)
   
The Dragon, Wapping, London (est. 1720)

The oldest appears to be that in Sandwich, which has been known as The Dragon since the sixteenth century, but was founded as St. George and the Dragon in 1446.

Given the reputation of Wapping in 1828, my guess is that's the one the poet had in mind.

Maybe.