I seem to recall that For the Honour of the Family had some similar themes to it - although that was set after the murder had already taken place, and the family were trying to figure out what to do afterwards.
One thing that seems to be common to just about every murder mystery show (and I'm sure your already aware of) is that EVERY character has some reason to lie to the police after the murder. Usually just because they're embarrassed, or what they've been doing is a minor offence, or it's actually relatively innocent, but in light of the murder it looks bad. Seems like that would be much of the basis for the web of intrigue to me. More or less, everybody has something they don't want the police finding out about. The more it potentially conflicts with the rest of the players, the better.
I do agree with you that it would potentially be on the nastier sides of family drama, which people tend to get enough of in real life (often more than they would like). I'm not sure how to deal with that, beyond perhaps giving a few major common talking points, so that the bickering doesn't necessarily have to come up.