I've also had a chance to break a Lee pin: as always when I did not have the spare part !!!
I saw carefully and I noticed that the pin looked glued. I tried to pull it out with a pair of pliers, but it did not come out; but when I warmed up JUST with a lighter, he went out alone.
I cleaned the seat with a drill bit of the size that came in with a very light resistance. then I broke the tip to the correct size and I fixed the cylindrical part (the one that is held by the drill chuck) with a bit of Attack! The drill bits (HSS ???) are much more resistant than the pin! I still keep drill bits as spare parts, but I've never used them!
I do not remember the measure: in Italy we have the decimal system and not in inch. I think they were tips of 1.4 mm.
In any case, just try possibly bringing the piece without pins into a hardware store.