Your guess is correct; "How lucky it is that the cat is a black one" is an accurate restatement of the phrase. "good job" is a particularly British term for "lucky".
So that leaves the question: why would it be lucky that the cat is black?
In all likelihood, this has to do with the tradition of wearing black clothes to a funeral or when mourning. There's a long history of wearing black while mourning in the English-speaking world; rumor has it that the tradition can be traced back to the Romans wearing black togas to funerals.
In any case, the mourning family is discussing what's required. The mention of black clothes would have come up. The child, not fully grasping that the cat doesn't necessarily wear clothes or attend funerals, remarks that it's lucky that the cat is already black, which saves the trouble of getting black clothes (or dye) for the cat.