The only thing missing is the OP's having been exposed to the double ended cables of many years ago. Yes, the inner cable has both ends capped with a lead/zinc end. the casing and adjuster are already installed on the inner cable. The likelihood of finding said cable is small. Both the length overall and the difference of length between the inner and casing must match up. I haven't had to deal with one of these for a long time.

The easy solution is to replace the caliper with a "modern" one that uses the anchor bolt type of cable securing. BTW modern freestyle bikes that have Orygs or Gryos use the same type of cable from their lever to the detangler but have far different shapes and lengths then what the OP's system needs. Andy.