Well, I am following Ben Lankford's plans for Model Shipways Constitution. He does not use the term "Burton Pendants" on these plans but seems to refer to them as "Running pendants" (Plan sheet #7 and Figure 8.1 in the instruction manual.) As I said in the original post, only the topmast pendants and their tackles are shown in Anatomy of a Ship series for the Constitution. There, these tackles are hooked to the pendants and attached to the channels.
Anyway, Lankford shows the pendants for both the topmasts and the lower masts but says nothing about how they are rigged to the tackles and the tackles attached to the hulls. In the "Masting and Rigging of English ships" it is stated that the pendants are usually not shown on static models. However, in the real ship, they must have been installed before the shrouds were if Lankford is to be believed. I have read that these pendants were simply tied to a shroud to prevent them flopping around.