I have eaten trout, hatchery brood stock that were dead in the water for 6-10 hours and never had a problem. They were in the cold water of the lake of course. Only time I have worried about a fish that has been dead for a long time is if they get mushy. If I feel the meat give/mush at all after it has been out for a while then I won't take a chance.
Is there any way to keep the live well colder? If you keep it at or below 55-60F then they should keep all day dead or alive, at least edible keep anyway.
Of course if it smells bad it is crab bait but I figured that was obvious.
I love taking my kids fishing, now if I could just manage to fish at the same time.