Catakig's author has replied. For some reason (in 1997) he chose colours that were HTML safe. This doesn't mean they're not good: he was eyeballing the composite output of an Apple II when he matched them up.
In lores order, and RGB range 0..5: 000, 401, 003, 405, 222, 024, 045, 210, 520, 222, 535, 050, 550, 153, 555
Re calculating the colours, I was going to mention that the "someone with good digital logic kung-fu" would also need black belts in NTSC and colour space math. Interesting then that Colin has also tried and rejected the very palette that MESS is using. ("Too bright," he says.)
Here's the lowdown from the seemingly knowledgable source, Robert Munafo: http://www.apple2.org.za/mirrors/ground.icaen.uiowa.edu/MiscInfo/Emulators/xapple2
Pics above have some colours closer to Colin's, and some to Rob's. And there are also European colours ...