my reccomendation for stripping the finish is to go to home depot, there's this stuff that's in an orange/copper colored bucket. it's called "EZ strip" or "Strip EZ", can't remember which one. Apply it with a plastic scraper, all over the body, toss it in a garbage bag for 1/2 an hour, then scrape all the goo off...
you can scrounge ebay for parts, or like you said, just buy another whole new system(this would be my choice) and keep the one you have now as parts. Since this thing isn't going to be holding its value(due to painting, and replacement parts) i'd just replace the pickup that doesn't work, and keep the other 2, unless you're not happy with the sound.
Should be a fun project though...good luck