The Liquidlogic Mist 9.5 (used to be the Heritage Featherlight) is one of the few kayaks that meets your requirements. I never paddled one and not sure if I would recommend it for you. But something like the Mist is what you are looking at with those specs.
Overall, most small plastic rec sit in kayaks around 10' long will claim weights between 40-50 lbs. Sit on tops usually weigh more then a similar sized sit in kayak. If you are willing to have a slightly heavier kayak you get a lot more choices.
When you say you want a min weight of 280lbs is that your weight or your weight plus the gear you are thinking of bringing. For a kayak people really need to know your weight and height to help make recos.
Since you have a black lab (me too) and children, I would wonder if you would be better off getting a canoe that you could take the dog and children with you as well as paddle solo. That would mean a lot longer than what you are thinking.
Kindle / iPad Cases
Paddler's Truck Rack
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