Is it acceptable or even allowed for the best in show judge to have one of his/her dogs being shown at the same show by another member of his/her family?

I'm not sure about allowed but it's certainly not ethical.   Should the dog take BOB, I'd assume his handler would withdraw him from BIS competition!

It is against the KC show regs for a judge to exhibit(or handle)a dog at a show. This rule lead to dogs being transferred over to partners etc so that they could be shown-the rules re showing dogs under a judge that bred them has been changed so that you can never judge a dog in breed that you have bred & the transfer action is now discouraged

Good (now discouraged - transferring a dog into another name for a specific show).   It happened in my breed once - it was a major show but honestly, if an owner/exibitor can't refrain for just one show, that's sad.