Best is still getting the entire harness, as @Big_John said. Good luck getting someone striping one to remove it for you, or even getting a proper pigtail.Is it possible to put the wires I currently have feeding the switch into a new plastic piece? Rebuild the replacement connector with the existing wiring instead of splicing a pigtail? (I would totally still have them send a 2' pigtail though, that just seems like a smart idea)
You could just use the shell, as you mentioned, but you will need to carefully inspect each terminal... that's where the heat came from, one of them is cooked... so you would still need to replace that one and the switch. Removing pins from the shell is a matter of patience and creativity. You are unlikely to find the proper tool, which may have never existed in real world anyway. Picks, paperclips, needles and improper use of existing terminal tools is the standard operating procedure for this job. Then you have a shell to find, a terminal to find and wire to inspect. A short pigtail would force you to this, but also that entire wire may need to be replaced if the entire circuit got hot (vs just the terminal). You really need to do an assessment once you're in there.
I'd get the harness or pigtail before starting.