Going from 2 bunks to 3 bunks limits your options. Going from 3 to 4 limits them further. That said there are some options that might work. My sister and her family has a lightweight quad bunkhouse that they pull with a 3.5 Expedition EL and it does pretty well. There is a trade-off - the queen bed is in a slide that pops out the back and it's obviously a lightly built camper and the interior space is limited with no side slides and the shorter length (I think its an Aerolite or something like that - there are other manufacturers with similar rear slide bunkhouse floorplans).
If you want a more standard type of TT bunkhouse I think you're going to struggle pretty mightily unless you step up to a HD SUV or pickup. Problem being on the SUV side of things your options are limited and getting increasingly old. Our pickup technically seats 6 but I wouldn't want to put any of our kids in that front center spot for very long, certainly not a trip that's 1000 miles each way. But for the shorter legs it might work. Also consider that running a minivan as a secondary vehicle on a 2000 mile trip would only run you an extra couple hundred bucks in fuel, which isn't all that much to add to a trip to Disney for a family of 6. That would open you up to HD pickups and then the world is your oyster on trailer/FW floorplans, and I think you'd find towing a large bunkhouse on a long trip more relaxing with a more capable tow vehicle. Just something to consider.
I'm not real knowledgeable about what options you might have on the motorhome side of things.
