w/ the mix of fish you describe, you probably dont need to cull any GSF, the LMB should take care of them depending on how many LMB you have.
the GSF are top predator in my lil sunfish pond, and they recruit at least twice in a season. when i cull, i remove all GSF under 8-inches, unless i judge a 6 or 7-incher showing top quality genetics/growth. right now i leave the BG alone.