I know you have specified a keelboat, however for camping with kids you may want to consider some swing keel / drop keel trailer sailers. I use to own a half share in a Farr 7500 which we left permanently in the water at a yacht club and only occasionally used the trailer. I regularly had 5 onboard and could pull the boat up right against the shore and the sprogs could get off anytime they liked and play on the beach. With the poptop and canvas cover the cockpit can be a useable extension of the living space.
The Farr 7500 would be within your price range, plenty of other also would.
Pictures say a 1000 words, below is what I mean