I think the problem is that Disney classifies King bed rooms as holding a MAXIMUM of 2 people. They will not let you book a king room with 3 people on your reservation, unless you choose a suite.
There is a room type in the garden wing of the Contemporary. It is not too expensive, and it has a king bed and a day bed in it. It will accommodate 3. It is classified as "Garden Wing, King". It is $320, $380, $420, $480/night in Value, Regular, Peak, and Holiday seasons, respectively. It's the best value if that is the room type you are looking for.
Otherwise, the 1 bedroom DVC units have what you want, but not the studios (those have a queen bed).
All the "1 bedroom suites" at WDW are one king bed and either a daybed or pull out sleeper sofa in the sitting area as well, and include concierge service, but the prices start around $500+ per night.