Going back to the original situation, petted dog on FILs lap, child piling on top - this is how puppies play, and they nip each other often enough. As they grow, so this should be trained out of them. So presumably the dog was jealous of someone trying to oust him from his favoured position and was showing his rival who's boss. The dog and FIL learned nothing from the event!
Perhaps a few case studies would help to illustrate your concern to them. There was a spate of young children being attacked by 'the family dog', often when being looked after by grandparents. Dogs seem to go for the faces of children, causing extreme scarring and in some horrible cases, the child dies (and sometimes the grandparent goes to jail). You are absolutely not over-reacting to the situation - it's a potentially very serious one - and it is your duty as a parent to take all necessary steps to ensure your children's safety.
Sometimes you have to upset someone by giving them the truth, in this case that their behaviour falls short of safe, and that although you are delighted they come to stay and are involved in the care of your children, they do so under your rules. Sure, they'll take a huff, which should wear off in a while, but even if it doesn't, at least your children will be safe. Seriously if your in-laws truly take offense at this then the problem is theirs not yours.