I had a feeling you might not want to rebuild the section of house. When I rebuilt-ish Claire's house I did it in stages.
I was going to offer to help, but if it's the house I'm thinking it is, I'd have to have the shell of the house without the CC windows.
There are some players who have claimed to be able to place basement stairs inside a house with foundation with a build cheat. I've never gotten that to work.
That's why I was suggesting to basically rebuild the section of house you want the stairs to be in.