There seems to be some confusion about temps. Pure lead melts at ~ 327 C (Celsius) or
621 F. Any lead alloy will melt at a somewhat lower temp. While Pb doesn't boil until you get above to 1750 C much above the melting point of your alloy will increase the amount of oxidation and vaporization. Pb is a heavy metal and therefore toxic.
PID's are a great way to go if you want good temp control. I had one on a muffle furnace that would hold to about a degree C at 625.
Since you are trying to control the temp of the Pb I'd put the thermocouple in the melt!
Just my tuppence.
“I often say that when you can measure what you
are speaking about, and express it in numbers, you
know something about it; but when you cannot
measure it, when you cannot express it in numbers,
your knowledge is of a meagre and unsatisfactory
kind; it may be the beginning of knowledge, but you
have scarcely in your thoughts advanced to the
state of
Science
, whatever the matter may be.”
Lord Kelvin, 1883