Have you thought about Island Cruises? They are very good for a family cruise as are more relaxed and you can eat at any time to suit you. For us it meant a few nice lie ins and being able to forget about times and routines for a week. We still dressed up for dinner (smart casual that is) as we enjoy doing this and everyone else also did so. The Steakhouse which is one of the more formal places to eat was a nice treat and well worth the supplement. For us it is difficult to imagine how formal cruising and children can mix successfully.
Thomson sounds up his street.
Reading other reviews it would seem that Ocean Village would suit you - hopefully he does wear a shirt. Seriously though you should be advised that many cruise ships are smart in the evenings and this means a jacket, shirt and tie - otherwise you will probably feel a little out of place and even be excluded from some area of the ship. there are some cruise lines that advertise "casual only" but have never been on one.
How about trying Ocean Village? A great family cruising holiday without any formal nights. Wear what you like.
As a cruise consultant and cruiser I suggest you look at Norwegian Cruise Line with freestyle dining and Princess
on certain ships offer personal choice dining. Also on most cruise lines there are alternative dining choices if you wish to avoid the Formal evenings (the amount of these formal evenings varies depending on the length of your cruise.
Why don't you look at Norwegian Cruise Line, they have a much less formal approach to cruising and much greater freedom with dining arrangements.
Island Star is informal and I've just come off Royal Caribbean Liberty of the Sea with my two kids 11 and 12 they had a great time and although in one week there are two formal evenings you can go off to Windjammer Restaurant wearing casual clothes. We dressed up and enjoyed it but it is for you to choose.