I am in the middle of a study to see how big CC need to be to eat Gshiners (conducted under the auspices of Prof. Cody). The biggest CC I have pulled out of my pond and measured since starting the study was 22" long, weight in the 4 to 5 lbs range, and had a max. mouth opening of 2.25".
How tall is your 6" long BG? I estimate it would be 3" or more tall (using Condello's current pictures of 7 inchers for source material), NOT counting the fins (we'll give the CC credit for swallowing headfirst and talking the BG into not extending any spiny fins).
I have some CC in the pond up to 6 lbs or so now, and will be feeding them to increase their size; I will report my findings here when I have caught a few more, including at least a couple of my biggest ones. But I think it's clear that it would take a BIG CC to consume a medium-sized BG.
Personally, if I wanted CC as the only predator in a small pond and also wanted BG, I would go with Hybrid BG. I think that is one application where HBG would be called for - the CC should be able to handle the relatively small numbers of F2 offspring the HBG would produce. I don't think CC alone can keep up with regular BG reproduction. Perhaps if you made aggressive use of traps and seining to remove small BG, you could make it work.