id blame cap(s) and resister(s) for the heat issues..
Also who knows what your connecting to with the solder repair.
id probably get down and dirty and repair all the damage replace caps, and resisters.
for SMD i would recoomend using a hot air station, and flux with solder in it. "just dab a bit where you want the solder to be put the smd item on the area with tweezers and then heat it up with the hot air when you have it placed properly.
the wire repair i would do with some kaynar wire only expose the smallest ammount of wire possible then tin that up as you tin that up more wire will get exposed. make sure you dont expose to much.
use less solder than you have been you only need a tiny ammount. use a fiber glass pen to expose the traces you need to work on. try and protect any thing you accidentaly exposed by using laquer or even nail varnish.
seems like its by passing some caps and resisters to me though. so i would definatly put a resister inline after calculating what exactly should be coming out the other end.
honestly seems like a lot of work. but if you dont want to/cant RGH then its worth the effort. if only to be able to extract the key,