I agree you're making a very generous offer, but I would try to scale it back to what you could afford for everyone--as others suggested, maybe providing accommodations and tickets for everyone, and letting them pay for their own airfare and meals, for example. If Casey ended up asking her parents for help paying for that part, fine, none of your business how the money got there. But I think it would be weird to go around Casey--you may feel you know her financial situation backwards and forwards, but I think there's the potential for offense when one presumes someone else can't pay for something, you know?
If it were me I would take what I think I know about Casey's financial situation (and anyone else's except my own) out of the equation, and just offer to pay for what I want to and what I know I can afford. I think it's a good impulse to say, "They can't afford stuff like this, so I will offer to treat them," but I think it can sometimes lead to stickiness down the line, so I would rather approach it as, "I can afford this, and it would make me happy to treat them."
Also, I would ask John and Samantha how they feel about including the two extra kids on the trip. I don't know if you've mentioned the trip to them at all yet, but it would definitely be a different dynamic with two older kids there, and that may or may not be something they're comfortable with. Once you have their okay and a firm plan about what you'll be paying for and what the others will be responsible for (with numbers), I would approach Casey away from the kids and put the plan to her, maybe in email or something so she has some time to think it over. There could be a lot of other considerations besides money for her, such as the dates you plan to be gone, something special Casey wants to do with her kids herself sometime, her feelings about the kids going to WDW in general, etc..
I think it is a really great impulse on your part, but I would just approach it carefully, because someone could hear your generous sentiment as "you can't afford to do stuff for your kids so we're going to fix that" and be hurt by it.