If the dogs and kids won't be all into your stuff, a single-head mini-split with the head in the open kitchen/family area may be able to handle it all if you leave the doors open.
With that approach there may be some room-to-room temperature differences, some of which is easy to mitigate, others less so, but no matter what the mini-split would still dry the air and provide at least partial cooling in the other rooms.
If the bedrooms or other doored-off rooms are on the south or southwest side with a lot of afternoon solar gain, a 2 or 3- head multi-split might be called for.
Is there any exterior wall insulation (particularly in the doored-off areas)?
Window types, and orientation?
Are the ducts in the attic, above the insulation? (If yes, is there at least a radiant barrier between the ducts and roof deck, and are the ducts sealed & insulated?)
What is the cooling BTUs/tonnage of the existing system?
All of these will affect the decision of whether to go ductless and the relative sizing of the unit compared to the previous system. There may be combination of envelope upgrades (eg. blowing insulation into uninsulated wall cavities, adding more insulation in the attic, putting up low-E storm windows, etc.) that may both enhance comfort in the rooms that don't have a separate zone head, evening-out the room-to-room temperature differences, as well as saving on cooling & heating (and up-front unit) costs.
And yes, continuously-variable inverter-drive compressors send the part-load efficiency of these things SOARING over traditional ducted heat pump systems. While there are inverter drive ducted systems out there, they're pricey and still fall short on ductless for efficiency for several reasons, such as duct leakage, direct duct gains/losses when routed outside of conditioned space, higher power needed to move air via ducts- it all adds up.
For weather data & looking up outside design temps, what's your zip code?