As far as I know, the JDM two-bulb corners are practically the same thing as the U.S. versions, which also are constant on with the headlights. They switched to the single bulb corners for use as blinkers when the projector fogs came out (the old fogs housed the blinkers). At that point, since the corner marker was now used as a blinker, they added "city lights" to the headlights for use as corner markers.
Even wired together, those small bulbs in the two-bulb corners shouldn't get hot enough to melt things. I would look for the problem elsewhere.