Hi again,
Please post a pic (close-up) of the 12" rack tom. 22" or 20" bass drums are both common for those set-ups (12"/13"/16" toms). I think that one of the early pre-owners had to solve a problem with a cracked original 12" tom and he was lucky to find somehow a fitting (= the one with the same tom bracket) 12" Tama tom in acryl. That's maybe how the Tama tom came to your Star set. In your case I don't believe that the original set-up was just a 3 piece configuration.
Ralf
NB: I guess you have to buy a fitting tom with the lugs you're looking for (either from Star or an early Tama one) on ebay to get them. That type of single toms appear either in USA or Europe (preferably Germany, Belgium/ Netherlands or Switzerland - and maybe Scandinavia as well?).