To solve the problem of heat inside the cone (or any other sensitive area) you could use a resistance soldering tool. As opposed to normal soldering tools, resistance soldering tools use two high-amperage electrical terminals in the tip that are bridged by the solder, melting it onto the joint. This method controls and concentrates the heat very effectively in relation to conventional soldering tools. Just don't poke yourself with the tip. You'll probably get a worse burn than even a normal soldering pen.
Resistance soldering tool is a crappy term for these things, since normal devices use resistance too, but to heat the tip itself rather than just what you're working on.
This is principle behind the Cold Heat thing you always see on TV (which, imho, is a piece of ****).