Thanks all for your comments. There are a lot of good ideas to share with the family.
It sounds like the beach barbeques are within our (and our kids') budget. We will hit at least one of them.
On our previous trips, without the kids, my wife reminded me that she was on VACATION and did NOT want to cook dinners. Hot dogs in the galley and chicken on the grill were my responsibility. With the kids along, we plan to eat half(?) of our dinners aboard. We do plan to split up for at least a couple dinners for budgetary reasons and just to get a break from each other.
We haven't settled on a menu that meets everyone's taste and dietary preferences yet. When we get that out of the way, we will choose whether to bring our food, pre-order provisions, or buy as we go. I may be back with more questions later. :-)