I always laugh when someone calls a trip that's a couple months away "last minute" as we almost never book our trips further out than that. The fact that we currently have a trip booked for 10/29 is highly unusual for us. We typically book 6-8 weeks in advance.
Four of my last five Disney World trips were literally booked 6-12 hours before I arrived at the hotel. For folks living in Florida, that sort of trip isn't as unusual as people think. Every Friday night, the Disney Reservations Center (where I used to work) gets slammed with calls from people looking for a hotel room for that night.
Obviously, coming from Europe is a different animal.
Thank you all for your suggestions. Looks like we will plump for the Waldorf Astoria.
Looking at your dates, I'd say the Waldorf Astoria is your best option. The only Disney Resorts under $400/night are moderate or value. Those will be a huge disappointment if you're hoping for luxury.
Once you add in the resort fee, the Waldorf Astoria is about $300/night plus tax. Or, it's $246/night if you're bold enough to book a non-refundable/non-changeable reservation. (Personally, I'd avoid a non-refundable reservation unless you have great travel insurance.)
Although not on Disney property, you might consider the Gaylord Palms. It's more upscale than luxury, but the building is unique in that it has several massive atria. It's a large hotel with convention facilities, so you may see people with name tags in business casual attire. But, that means fewer kids, too.
For your dates, a room facing the atrium is $266/night. The rate has 3-day cancelation policy. For this hotel, I'd pay the extra for a room on the atrium. No matter the weather, you can still enjoy the balcony because you're inside.
Link to the Gaylord Palms (which is a Marriott brand):
https://www.marriott.com/hotels/travel/mcogp-gaylord-palms-resort-and-convention-center/
Example of one of the three atria...this is the Everglades Atrium:
