I got the lock fixed tonight; ordered replacement parts through an eBay seller, and they were quick enough shipping the parts, but one of the parts that came was defective, and the shape of the spring that came with the repair kit was just slightly wrong. Thankfully it was a different part that was broken, so I was able to reuse the original pieces.
In case anyone else runs into this problem and has to order a lock cylinder repair kit, this is the best video on the repair I was able to find:
BMW E46 Door Key Lock Repair DIY - YouTube