IME large CC (up to about 26" or so in my pond) don't go after BG at feeding time, they concentrate on the feed (unlike LMB, which can often be seen going after feeding BG). The shear SIZE of them may be somewhat of a deterrent to BG; I spread the feed out over a large area to give the BG plenty of CC-free water to eat in.
BG seem to do OK on lower protein feed when it is a supplement to a natural diet. If it was all that they had to eat, I'd want it to be the best feed I could get, but as a supplement, stuff in the 28-32% protein range is probably OK.
Last year I fed three dozen BG in a tank 32% Purina Game Fish Chow (with a small weekly bonus of worms) for over 2 months without any apparent problems.