I know you said not to.. but seriously, stop being concerned about this.

really

Just get a vacuum desoldering tool at radio shaft and suck up all the leaded solder with that. Do it outside if you must but just do it.

With mine at least, the removed solder can be ejected after sucking it up so just get a little tin can or something to dump the old solder into.

Braid is another option that you might like more since you can just snip off the ..infected/contaminated.. section. However it is more exposed in general and you might not like that part of it.

This level of lead is not gonna do harm. If you must you can use gloves and a mask. But just suck it up and do it.