When you say your using soft lead where are you getting your lead ? It sounds like that's your problem. If your lead is pouring out in a steady flow and your mold will heat up after a few shoots then the only thing it can be is your lead is to hard. Small details need a very soft lead to fill.
lead ingots for pouring cast iron joints in plumbing pipe( from my supply house) as far as I know it's pure lead is scratches easy enough. I don't have a bottom pour furnace , just a cast iron pot and a laddle . My molds are warm but not burn your hand hot might have to try warming up hooks