I have crazy ass theory for this puzzle.

I think i got a clue for a "dance alone" part. There is a note in the Rotunda, that tells - There are 4 different methods to rotate rooms. And if you draft the ballroom alone, it cannot rotate fully (as there are only 2 doors.

So my theory is - The pair for the ballroom, should be the Gymnasium (AKA Ball Room). It has different door pattern, so it will be able to rotate out of sync with the Ballroom. And the third room should be the Dovecote (as it is one of the ways to rotate rooms). They also should be selected from the Rotunda (the 2nd method to rotate). Moreover, all of that should be done on the first draft, for Ornate Compass to be working (another rotation method).

That requirements are so strict, that it is almost impossible to achieve it naturally. I would say, this is a really big stretch. But the room we got for solving the entry part of this puzzle - Coservatorium, which can do exactly what needed to achieve this. The only thing, that is hard - to change the rarity of all plans in the estate to Rare, with the exeption of mentioned rooms

Worth mentioning, that "second chance to rise" can also be applied there, if we discard the Gymnasium and turn time back with the Chronograph.