I've done similar work, though I'm in Scotland so that probably isn't any use. I've never worked on this exact style of pickup, but I've replaced hookup wire on a few pickups, including a '70s Fender WRHB. This required new coax cable to be spliced onto the original wire within the pickup housing, and the shield of the coax to be soldered to the baseplate of the pickup.
Honestly it isn't scary if you don't have to go into the pickup coils themselves. You want a variable temperature soldering iron so that you can work hot enough to solder the shield to the pickup base, but reduce the temperature enough that you don't melt the insulation everywhere.
