Back in 2014, we booked the Citricos Chef's Domain table (I believe this is now the Sommelier Room) for my parents' 50th wedding anniversary. The max number of guests used to be 12 - I suspect it's decreased now - and we were able to select items from the standard menu, with a minimum overall spend. It was NP to get to the minimum with wine. We had a wonderful experience. We ruled out some of the other rooms at Disney-run restaurants listed above (e.g. Napa and Sonoma) because they required an event/catering menu, which was extreme overkill for our group. Jiko had a private room as well, but it was larger with larger minimum spend, and there were some other issues that escape my memory - doesn't matter at the moment, though, since they're still closed.
A couple of years ago, this topic came up, someone wanting to do a private party for 10-12 or so without the fuss of an event menu; there were some excellent suggestions for Disney Springs-area restaurants, which are often lovely and far less restrictive. Some great ideas above for off-site and Springs restaurants, and I'd also recommend calling any other places that appeal to your group at Disney Springs or maybe Swolphin to inquire about possibilities. Good luck!
(ETA - all of the above required specific emails/phone calls - it will take more effort than an ADR, but if you can call the restaurant directly, that is generally the easiest route.)